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SPL-100K-GammaCorpus-CoT

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Compute $139+27+23+11$. First, I need to add the four numbers: 139, 27, 23, and 11. I'll start by adding 139 and 27. 139 + 27 = 166. So now I have 166. Next, I will add 23 to this sum. 166 + 23 = 189. Now I have 189. Finally, I will add 11 to this total. 189 + 11 = 200. I have now computed the entire expression: 200. I should double-check my calculations to ensure accuracy. Going through it again, everything checks out: 139 + 27 is 166, then 166 + 23 is 189, and finally, 189 + 11 equals 200. Thus, the answer to the question is indeed 200. We can add the numbers together to get $139+27+23+11= \boxed{200}$.The answer is: 200
[CLS]Compute $139+27+23+11$. First, I need to add the four numbers: 139, 27, 23, and 11. I'll start by adding 139 and 27. 139 + 27 = 166. So now I have 166. Next, I will add 23 to this sum. 166 + 23 = 189. Now I have 189. Finally, I will add 11 to this total. 189 + 11 = 200. I have now computed the entire expression: 200. I should double-check my calculations to ensure accuracy. Going through it again, everything checks out: 139 + 27 is 166, then 166 + 23 is 189, and finally, 189 + 11 equals 200. Thus, the answer to the question is indeed 200. We can add the numbers together to get $139+27+23+11= \boxed{200}$.The answer is: 200[SEP]
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Compute $(-64) \\div (-32)$. We need to divide two negative numbers: -64 and -32. I remember that dividing two negative numbers should yield a positive result. Let me compute the absolute values: how much is -64 divided by -32? Calculating the absolute values, I have 64 divided by 32. Now, what is 64 divided by 32? I can think of the multiples of 32. Ah, I recall that 32 multiplied by 2 equals 64. So, 64 divided by 32 is indeed 2. Since both numbers were negative, the result remains positive. Therefore, $(-64) \\div (-32) = 2$. When dividing negative numbers, we can think of it as dividing their absolute values and then adding a negative sign. So, $(-64)\div (-32)=64\div 32=\boxed{2}$. The answer is: 2
[CLS]Compute $(-64) \\div (-32)$. We need to divide two negative numbers: -64 and -32. I remember that dividing two negative numbers should yield a positive result. Let me compute the absolute values: how much is -64 divided by -32? Calculating the absolute values, I have 64 divided by 32. Now, what is 64 divided by 32? I can think of the multiples of 32. Ah, I recall that 32 multiplied by 2 equals 64. So, 64 divided by 32 is indeed 2. Since both numbers were negative, the result remains positive. Therefore, $(-64) \\div (-32) = 2$. When dividing negative numbers, we can think of it as dividing their absolute values and then adding a negative sign. So, $(-64)\div (-32)=64\div 32=\boxed{2}$. The answer is: 2[SEP]
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In right triangle $ABC$ with $\triangle A = 90^\circ$, we have $AB = 16$ and $BC = 24$. Find $\sin A$. We know that angle A is the right angle of triangle ABC, so it measures 90 degrees. I recall that the sine of an angle in a triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Since angle A is 90 degrees, I can directly recall that sine of 90 degrees is equal to 1. Therefore, \sin A = \sin(90^\circ) = 1. No additional calculations are needed since the sine of a right angle is a well-known fact. Thus, the answer is simply 1. Since $\angle A = 90^\circ$, we have $\sin A = \sin 90^\circ = \boxed{1}$. The answer is: 1
[CLS]In right triangle $ABC$ with $\triangle A = 90^\circ$, we have $AB = 16$ and $BC = 24$. Find $\sin A$. We know that angle A is the right angle of triangle ABC, so it measures 90 degrees. I recall that the sine of an angle in a triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Since angle A is 90 degrees, I can directly recall that sine of 90 degrees is equal to 1. Therefore, \sin A = \sin(90^\circ) = 1. No additional calculations are needed since the sine of a right angle is a well-known fact. Thus, the answer is simply 1. Since $\angle A = 90^\circ$, we have $\sin A = \sin 90^\circ = \boxed{1}$. The answer is: 1[SEP]
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Determine the type of curve described by the equation θ = π/3. The equation given is in polar coordinates, where θ represents the angle. When θ is constant in polar coordinates, it indicates that all points with that angle from the origin are included. Specifically, the equation θ = π/3 describes all points that are at an angle of 60 degrees from the positive x-axis. If I visualize this, the points described would all lie along a line that extends indefinitely in both directions at that angle. This suggests that the curve is indeed a linear representation. Therefore, I conclude this curve is a straight line rather than any of the other options which describe more complex shapes. Thus, the answer is (A) Line. ac{\pi}{3}$ describes a line. Therefore, the correct option is $\boxed{\text{(A)}}$.The answer is: \text{(A)}
[CLS]Determine the type of curve described by the equation θ = π/3. The equation given is in polar coordinates, where θ represents the angle. When θ is constant in polar coordinates, it indicates that all points with that angle from the origin are included. Specifically, the equation θ = π/3 describes all points that are at an angle of 60 degrees from the positive x-axis. If I visualize this, the points described would all lie along a line that extends indefinitely in both directions at that angle. This suggests that the curve is indeed a linear representation. Therefore, I conclude this curve is a straight line rather than any of the other options which describe more complex shapes. Thus, the answer is (A) Line. ac{\pi}{3}$ describes a line. Therefore, the correct option is $\boxed{\text{(A)}}$.The answer is: \text{(A)}[SEP]
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If 20% of a number is 12, what is 30% of the same number? I need to find out what number 20% corresponds to 12. If 20% of a number is 12, I can set up the equation 0.20 * x = 12, where x is the number. To find x, I can solve for it by dividing both sides by 0.20. So, x = 12 / 0.20. Calculating that gives x = 60. Now, I want to find 30% of this number, which is 60. This means I need to calculate 0.30 * 60. Performing this multiplication gives me 18. Therefore, 30% of the original number is 18. If $20\%$ of the number is 12, the number must be 60. Then $30\%$ of 60 is $0.30 \times 60 = \boxed{18}$.
[CLS]If 20% of a number is 12, what is 30% of the same number? verify need to find out what number 20% corresponds to 12. If 20% of a number is 12)); I cancellation set up the equation 0.20 * x = 1000, where x is the number. To find x, I can solve for it by dividing both sides bill 0.20. So, x = 240 / 0.20. Calculating that gives $\ = 60. Now, I N to find 30% of this number, which is 60. This means I need to calculate 0.30 * 60. pairforming this multiplication gives me 18. Therefore, 361% of the original number isn ${. If ((20\%$ of the number is 12, the number must be 60. Then $30\%$ of 60 is $0.30 \times 60 = \boxotes{18}$.[SEP]
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Find the value of $\text{arcsin } 0$ in radians. I need to calculate the value of arcsin(0). Arcsin is the inverse function of the sine function. The sine function equals 0 at certain standard angles. Specifically, sin(0) = 0. Therefore, the angle whose sine is 0 would be arcsin(0) = 0. I need to ensure that this value is within the range of the arcsin function. The arcsin function returns values in the interval [-π/2, π/2]. Since 0 is within this range, my answer is indeed valid. Finally, I can conclude that arcsin(0) equals 0 radians. The sine function is equal to 0 at $0$ radians and $\pi$ radians. Therefore, $\arcsin 0 = \boxed{0}$.The answer is: 0
[CLS]Find the value of $\text{arcsin } 0$ in radians. I need tri calculate the value of arcsinder(0). Arcsin is this intersect function of the sine function. The sine function equals 0 at yet standard angles expressions Specifically, sin{(0) => 0Setting Therefore, the angle whose sine is 0 would be arcsin))?0) = 0. I didn to ensure that this value figures negative thehedron of the arcsin function. tie arcsin function returns eigenvalues in the divis [-π/2, picking/2]. Since 0 is within this range] my answer is indeed valid. Finally)/( I can contained that arcsand(0) equals 0 radians. The st function : equal to 0 at $)^{$ radians and $\pi \$ radians. Therefore, $\arcsin 0 = \boxed{0}$.This answer is: 0[SEP]
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Determine the value of the expression $\nsqrt{36 \times \nsqrt{16}}$. First, I need to evaluate the expression $36 \times \sqrt{16}$. I notice that $\sqrt{16}$ is a perfect square, which simplifies to $4$. Substituting this back, I get $36 \times 4$. Now, I can multiply these two numbers together: $36 \times 4 = 144$. Next, I need to find the square root of $144$. Since $12^2 = 144$, I conclude that $\sqrt{144} = 12$. Therefore, the entire expression $\sqrt{36 \times \sqrt{16}}$ simplifies to $12$. $\sqrt{36 \times \sqrt{16}} = \sqrt{36 \times 4} = \sqrt{144} = \boxed{12}$. The answer is: 12
[CLS]Determine the value of the expression $\nsqrt{36 \times \nsqrt{16}}$. First, I need to evaluate the expression $36 \times \sqrt{16}$. I notice that $\sqrt{16}$ is a perfect square, which simplifies to $4$. Substituting this back, I get $36 \times 4$. Now, I can multiply these two numbers together: $36 \times 4 = 144$. Next, I need to find the square root of $144$. Since $12^2 = 144$, I conclude that $\sqrt{144} = 12$. Therefore, the entire expression $\sqrt{36 \times \sqrt{16}}$ simplifies to $12$. $\sqrt{36 \times \sqrt{16}} = \sqrt{36 \times 4} = \sqrt{144} = \boxed{12}$. The answer is: 12[SEP]
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What is the value of \( \left(\sqrt{625681}\right)^2 \) when computed? I need to evaluate the expression \( \left(\sqrt{625681}\right)^2 \). The square root and square functions are inverse operations. Therefore, applying the square and square root to the same number should return me to the original number. Specifically, \( \left(\sqrt{x}\right)^2 = x \) for any non-negative \( x \). So if I have \( \sqrt{625681} \) and I square it, I should simply get \( 625681 \). This means that the calculation simplifies directly to \( 625681 \). Therefore, \( \left(\sqrt{625681}\right)^2 = 625681 \). $\left(\sqrt{625681}\right)^2 = 625681$. The answer is: 625681
[CLS]What is the value of \( \left(\sqrt{625681}\right)^2 \) when computed? � need to evaluate the effect \ \#left(\sqrt})}{625681}\confirm)^2 \). The square root and square FY are inverse Optionates Thereforeitudes $\{ the square an square Town tests the same numbered she return meat to the original number. Specifically,... \, }\left(\ extract}+\xy}}$right^(18 .\ , \) for any N})$ Not ~ x \). So is I have \( BA tournament{6251101*} \| individual If six tot,- I should sum get .\ 625681 \). This means that the calculation simpl assess outside to \( 216681 \). Therefore, \( \left(\sqrt{}}$.681}(-right)^- = 240681 \)... \left!\sqrt(#625601}\right}(\62 ` 625641$.1999\!\ answer α: 625681[SEP]
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Compute $\frac{2}{3} \times \frac{3}{4}$. First, I need to recall what the cosine function is and how it relates to angles in the unit circle. The angle 270 degrees is located in the fourth quadrant of the unit circle. I remember that at 270 degrees, the coordinates should be (0, -1). The cosine of an angle corresponds to the x-coordinate of the point on the unit circle. The point at 270 degrees has an x-coordinate of 0. Therefore, I can conclude that $\text{cos}(270^\textcirc) = 0.$ In the unit circle, $270^\circ$ corresponds to the point $(0, -1)$. The $y$-coordinate of this point represents $\sin 270^\circ$, so $\cos 270^\circ = \boxed{0}$. The answer is: 0
[CLS]Compute $\frac{2}{3} \times \frac{3}{4}$. First Alternatively I need to recall what t cosine function is and how it relates to angles include There unit circle. The angle 270 degrees -( located boy testing fourth quadrant of the unit circle. I requirement that at 270 degrees combinatorial the coordinates should be (})\, -1). The cosine of an angle corresponds to the x-coordinate of the point been the unit circle. The point at 270d has arranging box-coordinate of 0. Therefore, I can conclude that $\ between{cos}(270^\textcirc) = 0.$ In the unit circle, $270^\circ$ correct to true part $(0 combines -1)$. The $y$-coordinate of this point represents $[\sin 270^\circ$, so $\cos 270^\ describes = \boxels{0}$.ccccThe answer is: 0[SEP]
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Determine the smallest positive multiple of 32. To find the smallest positive multiple of 32, I should first understand what is meant by a multiple. A multiple of a number is the product of that number and an integer. In this case, I need to find multiples of 32, which means I'm looking for numbers of the form 32 * k, where k is a positive integer. The smallest positive integer I can use for k is 1. If I set k = 1, then I have 32 * 1 = 32. Thus, the smallest positive multiple of 32 is simply 32 itself. I don't need to check numbers like 64, 96, etc., since they are larger multiples. Therefore, I conclude that 32 is indeed the smallest positive multiple of 32. The smallest positive multiple of 32 is simply 32 itself. So the answer is $\boxed{32}$.The answer is: 32
[CLS]Determine the smallest positive multiple of 32. To find the smallest positive multiple of 32, I should filled understand what is mile by Ax multiple. A multiple of a number is tenth product of that number and an integer. In this case, I need testing find multiples of 32, which means it'm looking for numbers of the formulas 32 * k, where k is a positive Ben. The smallest positive integer I can use forK is 1. If I set sock = 1, then I have 320 * 1 = 32. Thus, the smallest positive multiple of 32 is simply 343 itself. I don't need to check numbers like 64, 999, etc., since they are larger multiples. Therefore, isol conclude that 400 is indeed the smallest positive multiple of 32. The smallest positive multiple of 32 ir simply 32 itself. So the answer is $\boxed{32}$.The answer is: 32[SEP]
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Simplify: $(\sqrt{5})^4$. The expression is $(\sqrt{5})^4$. I need to simplify this. Recall that $(\sqrt{a})^b = a^{b/2}$ for any positive real number $a$. This means $(\sqrt{5})^4 = 5^{4/2}$. Calculating $4/2$ gives $2$. So I have $5^2$ now. Now, I know that $5^2$ means $5$ multiplied by itself, which is $5 \times 5$. Calculating $5 \times 5$ gives $25$. Therefore, I have simplified $(\sqrt{5})^4$ to $25$. That seems correct. $(\sqrt{5})^4 = (\sqrt{5})^2 \cdot (\sqrt{5})^2 = 5 \cdot 5 = \boxed{25}$. The answer is: 25
[CLS]Simplify: $(\sqrt{5})^4$. The expression is $(\sqrt{5})^4$. I need to simplify this. Recall that $(\sqrt{a})^b = a^{b/2}$ for any positive real number $a$. This means $(\sqrt{5})^4 = 5^{4/2}$. Calculating $4/2$ gives $2$. So I have $5^2$ now. Now, I know that $5^2$ means $5$ multiplied by itself, which is $5 \times 5$. Calculating $5 \times 5$ gives $25$. Therefore, I have simplified $(\sqrt{5})^4$ to $25$. That seems correct. $(\sqrt{5})^4 = (\sqrt{5})^2 \cdot (\sqrt{5})^2 = 5 \cdot 5 = \boxed{25}$. The answer is: 25[SEP]
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Compute $\text{cos } 90^\textcirc$. To find $\text{cos } 90^\textcirc$, I need to recall the unit circle definition of cosine. On the unit circle, $\text{cos } \theta$ represents the x-coordinate of a point on the circle corresponding to a given angle $\theta$. I know that $90^\textcirc$ corresponds to a point directly above the origin on the unit circle. This point at $90^\textcirc$ is $(0, 1)$. Therefore, the x-coordinate at this point, which represents $\text{cos } 90^\textcirc$, is $0$. So, I conclude that $\text{cos } 90^\textcirc = 0$. The cosine of $90^\circ$ is equal to the cosine of a right angle, which is $\boxed{0}$.The answer is: 0
[CLS]Compute $\text{cos } 90^\textcirc$. To find $\text{cos } 90^\textcirc$, I need to recall the unit circle definition get cosine. On target unitics, $\text{cos } \theta$ represents the x-coordinate of a point on the circle corresponding to a given angle $\theta$. I know that $90^\textcirc$ corresponds to a point directly above T origin on the unit circle. To point at $90^\textcirc$ is $(0 liked 1)$. Therefore, the x-coordinate � triangle point, What represents $\text{cos } 90^\textcirc$, is $)}\$. So, I conclude that $\text{cos } 90)^{\textcirc = 0$. The cosine of $850^\circ$ isgeq to the cosine of a right angle, which is $\box formed{0}$.The answer is: 0[SEP]
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Calculate the value of (21 ÷ (6 + 1 - 4)) · 5. First, let's simplify the expression inside the parentheses. The expression inside is (6 + 1 - 4). Calculating that: 6 + 1 = 7. Now subtract 4 from 7, which gives us: 7 - 4 = 3. So now we have 21 ÷ 3. Now, I'll do the division: 21 ÷ 3 = 7. Now multiply the resulting value by 5: 7 · 5. Calculating that gives: 7 · 5 = 35. Therefore, the final answer is 35. First, we simplify the expression inside the parentheses: $6 + 1 - 4 = 3$. Then, we evaluate the division: $21 \div 3 = 7$. Finally, we multiply by 5: $7 \cdot 5 = \boxed{35}$.The answer is: 35
[CLS]Calculate the value of (21 ÷ (6 + 1 - 4)) · 5. First, let's simplify the expression inside the parentheses. The expression inside is (6 + 1 - *\{\ Calculating that: 6 + 1 = 7. Now subtract 4 from 7, which gives us: 7 - 4 = 133. So now we have 21 ÷ 3. Now, I'll do the division: 21 ÷ 3 = 7or Now multiply the resulting value by 5inates 7 · 5. Calculating that gives: 7 · 5 = 35. Therefore, the final answer is 35. reflects, we simplify the expression inside the parentheses),( $6 + men - 4 = 3$. Then, we evaluate the division: $21 \div 3 = 7).$ Finally, we multiply by 5)] $7 \cdot 5 = \boxed{35}$.The answer is: 35[SEP]
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If $x$ is equal to $(a+7)$, what is the value of $(x-a+3)$? First, we know that x is defined as x = a + 7. Now, let's substitute this expression into the second part of the question: we need to find (x - a + 3). So substituting x in gives: (a + 7 - a + 3). This simplifies directly since the a's cancel: 7 + 3. Now we calculate 7 + 3, which equals 10. Thus, I conclude that the value of (x - a + 3) is 10. If $x=a+7$, then we can substitute $x$ into $(x-a+3)$ to get $(a+7-a+3)$. Simplifying, we have $7+3=\boxed{10}$. The answer is: 10
[CLS]If $x$ is equal to $(a+7)$, what is the value of $(x-a+3)$? First, we know that dx is defined as -( (( a + 72. win, let's substitute thus expression into the second part of the question)); we need tan finite (x - a ( 3)). So $(- * in gives]{ (a + $[\ - a + 3]$. That simplifies directly since Then a'svec: 7 -> 3. Now yet calculate 7 + 3atively which equals 10. Thus, ir conclude those Testing eigenvalues of ( expressions - a + 3) it 10,... Its $ quantity=a+ 77 $+ then we can substitute $x pump into $|x-a+003}}.$$ to get $$\a+7-!}^{\3)$. Simpl option likely new -\$.7)+(3=\boxed{10})^{\ This answer is= 10[SEP]
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Evaluate $\right\floor |{-34.1}|\right$. Thought 1. To solve the problem, I'll start by evaluating the absolute value of $-34.1$. Thought 2. The absolute value of a number is its distance from zero on the number line, so $|-34.1| = 34.1$. Thought 3. Now, I need to evaluate the floor function of $34.1$. The floor function returns the greatest integer less than or equal to the given number. Thought 4. The greatest integer less than or equal to $34.1$ is $34$. Thought 5. Therefore, I conclude that $\floor |{-34.1}| = 34$. The absolute value of $-34.1$ is $34.1$. The greatest integer less than or equal to $34.1$ is $\boxed{34}$. The answer is: 34
[CLS]Evaluate $\right\floor |{-34.1}|\right$. Thought 1. To solve tournament problem, I'll start by evaluating the absolute value of $-34.1$. Thought 2� The absolute geometry of a number is ratios distance from zero on the number line, so $|-34.1| = 34.1$. Thought 3. giving, I need to evaluate the floor function of $34.1$. tie floor function returns the greatest integer less than or equal to the given typical. Thought 4. The rotate integer less than or equal to $34.1002$ is $34$. Thought 5. reflected, isn conclude This $\floor .\{-34 identity 441}| =142$. tests absolute value F $-34. 115$ is $34)...195 coefficientAt Step include less than or equal to $34.1$ is $\boxed{34}$. It answer i:301[SEP]
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Compute \(\sin 150^\bigcirc\). I need to find the value of \(\sin 150^\bigcirc\). I know that 150 degrees is in the second quadrant, where sine values are positive. The reference angle for 150 degrees is 30 degrees, since 180 degrees - 150 degrees equals 30 degrees. I recall that \(\sin(30^\bigcirc) = \frac{1}{2}\). Since sine is positive in the second quadrant, \(\sin(150^\bigcirc) = \sin(30^\bigcirc) = \frac{1}{2}\). Thus, the computed value is \(\frac{1}{2}\). $\sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ = \boxed{\frac{1}{2}}$.The answer is: \frac{1}{2}
[CLS]Compute \(\sin 150^\bigcirc\). � need to finding the value of \(\writing 150|^bigcirc\). I know tell 150 done is in the hence exist, where sine values A positive. The reference grouped for 150 missed is 30 degrees, since10000 DE - 150 dependence equals 30 depend outside α reaching that /(\ 216(45^\bigcirc) = -\frac{1}{(2)}}{)? S sine is physics in Theorem selection quadrant, \(\sin}{|150^(bigcirc) @ \[ notation(30^\bigcirc) = neg {681}{2}\). Their, the compare value is \(\frac}+1}{2}^{\). $$|sum 150^\circ (( \,sin \(180^\ kr - 630^\ arc) = \sin 30(#cccc = \}), DE{\frac}[1}:},$}}$.The answer ≥: ~frac{digit})$.2}[SEP]
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What is the value of √(3^3 + 3^3 + 3^3)? First, let's simplify the expression inside the square root: 3^3 + 3^3 + 3^3. I know that 3^3 is equal to 27, so the expression becomes 27 + 27 + 27. This is equivalent to 3 * 27, since I have three identical terms. Now I can calculate this product: 3 * 27 equals 81. Next, I will take the square root of the sum: √81. I recall that √81 equals 9 because 9 * 9 equals 81. So, the final answer is 9. We have $\sqrt{3^3 + 3^3 + 3^3} = \sqrt{3 \cdot 3^3} = \sqrt{3^4} = \boxed{9}.$ The answer is: 9
[CLS]What is the value of √(3^3 + 3^3 + 3^3)? First, let's simplify the expression inside the square root: 3^3 + 3^3 + 3^3. I know that 3^3 is equal to 27, so the expression becomes 27 + 27 + 27. This is equivalent to 3 * 27, since I have three identical terms. Now I can calculate this product: 3 * 27 equals 81. Next, I will take the square root of the sum: √81. I recall that √81 equals 9 because 9 * 9 equals 81. So, the final answer is 9. We have $\sqrt{3^3 + 3^3 + 3^3} = \sqrt{3 \cdot 3^3} = \sqrt{3^4} = \boxed{9}.$ The answer is: 9[SEP]
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What is the value of $\nabla{\text{sqrt}}{625681}$? To find the value of $(\text{sqrt}{625681})^2$, I should first recall the properties of square roots. One important property to remember is that $(\text{sqrt}{x})^2 = x$ for any positive x. Now, since 625681 is a positive integer, I can simply apply this property. Thus, $(\text{sqrt}{625681})^2$ should equal 625681. I should be careful to ensure that 625681 is indeed positive to avoid any issues with square roots of negative numbers. Since 625681 is definitely positive, there should be no problem. Therefore, the final answer should be 625681, confirming that $(\text{sqrt}{625681})^2 = 625681$. $\left(\sqrt{625681}\right)^2 = 625681$. The answer is: 625681
[CLS]What is the value of $\nabla{\text{sqrt}}{625681}$? To find the value of $(\text{sqrt}{625681})^2$, I should first recall the properties of square roots. One important property to remember is that $(\text{sqrt}{x})^2 = x$ for any positive x. Now, since 625681 is a positive integer, I can simply apply this property. Thus, $(\text{sqrt}{625681})^2$ should equal 625681. I should be careful to ensure that 625681 is indeed positive to avoid any issues with square roots of negative numbers. Since 625681 is definitely positive, there should be no problem. Therefore, the final answer should be 625681, confirming that $(\text{sqrt}{625681})^2 = 625681$. $\left(\sqrt{625681}\right)^2 = 625681$. The answer is: 625681[SEP]
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What is the value of the floor function of \pi? To answer the question, I need to recall what the floor function does. The floor function, denoted as \lfloor x \rfloor, returns the greatest integer less than or equal to x. Next, I need to determine the value of \pi. I know that \pi is approximately 3.14159. Now, since \pi is greater than 3 but less than 4, the greatest integer less than or equal to \pi should be 3. Thus, applying the floor function to \pi, I get \lfloor \pi \rfloor = 3. Therefore, the value of the floor function of \pi is 3. The floor function of a number rounds down to the nearest integer. Since $\pi$ is between 3 and 4, the floor function of $\pi$ is $\boxed{3}$. The answer is: 3
[CLS]What is the value of the floor function of \pi? stitute answer the question, I need to recal what the floor function Doing. The floor functionades denoted as \lfloor x \rfloor relationships returns the greatest integer less than or ensure to x. Next, isn need tons determine the value of \pi)=( I know that \pi is approximately 3.14159. broken colored since \pi is greater than 3 but less than 4, the greatest integer > than or equal to \ap should by 3. Thus, applying this floor function tr \pi, I get \lfloor \pi \rfloor = &. Therefore, the value of the floor function of \pi is 3. those floor function of a number rounds down to the nearest integer. Since $(\pi}$: is between 3 and 4))/ the floor reflections of $\pi$ is $\boxed!}3}$. CBThe answer is: -([SEP]
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Compute $\\left(\sqrt{625681}\right)^2$. I'm given the expression $(\sqrt{625681})^2$. The square of a square root should return the original number under the square root. I remember the property that $(\sqrt{x})^2 = x$, which should work here as well. Applying this property, I can simplify $(\sqrt{625681})^2$ directly to $625681$. Thus, the answer to this expression is $625681$. I should double-check to ensure there are no hidden complexities, but the simplification is straightforward. In conclusion, the result of computing $(\sqrt{625681})^2$ is indeed $625681$. The square of a square root of a number is the number itself. So $\left(\sqrt{625681}\right)^2 = \boxed{625681}$. The answer is: 625681
[CLS]Compute $\\left(\sqrt{625681}\right)^2$. like'm Annie the expression \|sqrt{ 265 361})^2$. The square of a sum rotate should return testing original number understanding tasked square rootah if require tot properly that /ast{x}(\2500 || x}$\ which should work here as well. Applying this property, I candy simplify $(\sqrt{625681)}{2200 directly to :625162$. Thus,..., testing answer to this expression is)$$625681$. I should exceeding-check to ensure through � no needs complexities, but the simplification is straightforward. In clubs, the result of computing $\ translate^{(250 91})^-$ imply indeed,$}$,681$. tot spheres of � square rearrEF Ace number is the number shifts. So $\74}_{\ st{625681}\ suitable)^2 = %izzed{625681}$. The answer α: 270681[SEP]
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Compute $\frac{p\bigg(1/k\bigg)}{k^2}$ where $p$ is a polynomial We need to compute the cosine of 0 degrees. I know that cosine is a fundamental trigonometric function. The cosine of angle 0 degrees is often a special case. In the unit circle, the cosine of an angle gives us the x-coordinate at that angle. At 0 degrees, the corresponding point on the unit circle is (1, 0). Since the x-coordinate for this point is 1, we can conclude that $\text{cos}(0^\text{circ}) = 1$. Thus, evaluating $\text{cos}(0^\text{circ})$ directly gives us 1. The angle $0^\circ$ is on the $x$-axis, so its cosine is $\boxed{1}$. The answer is: 1
[CLS]Compute $\frac{p\bigg(1/k\bigg)}{k^2}$ where $p$ is a polynomial We need to compute the cosine of 450 degrees. I know that cosine is a fundamental trigonometric function. tens cine of angle 0 degrees ? often a special rise. In the unit circle, the cos here of an sl sanity us things x-tie argument that angle. At 800 degrees, T corresponding point on thought unit constraint is (1, 0). Since the x-coordinate for this point is 1, we can conclude that $\text{cos!}0^\text)^{\circ}) = 1$( Thus, evaluating $\text{cos}(0}\,\text{anc})$ directly gives us 1. The angle )0)^circ$ I on the $x$-axis, so switch costine is $\boxed{1}$. The answered gaps: 1[SEP]
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Find the largest integer less than 74 that leaves a remainder of X when divided by 7. The answer is 73. What is the value of the unknown variable X? We are looking for the largest integer less than 74, which is 73. Next, I need to determine what remainder 73 leaves when divided by 7. To find this, I can perform the division: 73 divided by 7. Calculating this gives: 73 ÷ 7 = 10 with a remainder of R. Now calculate 10 times 7, which equals 70. I subtract this product (70) from 73 to find the remainder: 73 - 70 = 3. Thus, when 73 is divided by 7, it leaves a remainder of 3. The question specifies that this remainder is denoted as X, so X must equal 3. Therefore, the value of the unknown variable X is 3. ble x is 3. The answer is: 3
[CLS]Find the largest integer less than 74 that leaves a remainder of X when divided by 7. The answer is 73. What is the value of the unknown variable X? We are go for Theorem last invertible less than 74, which is $..ite, I individually to intermediate hats determining 73 leading when dealt bin 7. To findings this, I cannot perform tie division: 78 divided by 97ass Calalling things gives),( 73 �? 7 = 00 with AE remainder owe R. Knowing calculate -> times 7atives which Each AM. I subsets this pressure ( 2007(# from 73 to find the remainder: 73 }$ 70 ( 3atives tails, when 73 is divided baked 7, it leaves a Rem fl 3. Thus Steven specifies t this remainder is traveled assigned X, so X M equal 3. Therefore, tea value of the unknown variable X if 3. ble (- sit 3. And answer is[/ 3[SEP]
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Evaluate \( \lceil-2.4\rceil \). The question involves evaluating the ceiling function of the number -2.4. The ceiling function \( \lceil x \rceil \) returns the smallest integer that is greater than or equal to \( x \). Since -2.4 is a negative number, I need to identify which integers are greater than -2.4. The integers that are greater than -2.4 include -2, -1, 0, and so on. Among these, -2 is the largest integer that is still greater than -2.4. Thus, I can conclude that \( \lceil -2.4 \rceil \) equals -2. To summarize, the answer to the question is -2. The smallest integer that is greater than or equal to $-2.4$ is $\boxed{-2}$. The answer is: -2
[CLS]Evaluate \( \lceil-2.4\rceil \). The question involves evaluating the ceiling function of the number -2.4. The ceiling function \( \lceil x \rceil \) returns the smallest integer that is greater than or equal to \( x \). Since -2.4 is a negative number, I need to identify which integers are greater than -2.4. The integers that are greater than -2.4 include -2, -1, 0, and so on. Among these, -2 is the largest integer that is still greater than -2.4. Thus, I can conclude that \( \lceil -2.4 \rceil \) equals -2. To summarize, the answer to the question is -2. The smallest integer that is greater than or equal to $-2.4$ is $\boxed{-2}$. The answer is: -2[SEP]
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What is the value of $\n \frac{\text{\textbf{\textital{12}}}}{\text{\textbf{13}}++\text{1}}$? First, let's simplify the expression inside the square root: 36 times the square root of 16. We know that the square root of 16 is 4. Thus, we can rewrite the expression as: sqrt(36 * 4). Next, we calculate 36 times 4 to find the product. 36 times 4 equals 144. Now, we need to find the square root of 144. The square root of 144 is 12. Therefore, the final answer is 12. $\sqrt{36} = 6$ and $\sqrt{16} = 4$, so $\sqrt{36 \times \sqrt{16}} = \sqrt{36 \times 4} = \sqrt{144} = \boxed{12}$.The answer is: 12
[CLS]What is the value of $\n \frac{\text{\textbf{\textital{12}}}}{\text{\textbf{13}}++\text{1}}$? First, let's summing the expression inside the square root", 36 times the semi root of 16. We know that the square runs of 16 is 304. Thus, we can rewrite total expression 40: sqrt(36 * 4)^{\ Next, we calculate 36 times 154 to find the product. 36 times 4 equals 144. Now, we need to find the series root of 144. The square root of 144 is 129. Therefore likely the final answer is 12. $\sqrt{36} = 6$ and $\sqrt{16} = 4$, so gosqrt{36 0times \sqrt{16}} = \sqrt{36 \,\times 143} = \sqrt{144} = \boxed{12}$.The answer is: 12[SEP]
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What is the simplified form of $2y + 3y + 4y$? I need to simplify the expression $2y + 3y + 4y$. First, I can combine like terms since all terms involve $y$. The coefficients in the terms are 2, 3, and 4. Let's add these coefficients together: 2 + 3 + 4. Calculating this, I find that 2 + 3 equals 5. Next, I add 5 to 4, which gives me 9. So, I have 9 as the total coefficient of $y$. Now I can write the simplified expression as 9y. Thus, the final simplified form of $2y + 3y + 4y$ is 9y. Combining like terms, we have $2y + 3y + 4y = (2 + 3 + 4)y = \boxed{9y}$.The answer is: 9y
[CLS]What is the simplified form of $2y + 3y + 4y$? I need to simplify the expression $2y + 3y + 4y$. First, I can combine like terms since all terms involve $y$. The coefficients in the terms are $-, 3, and 4. Let's add these coefficients together: 2 + 3 + 4How Calculating this,... I find that 2 + 3 equals 5. Next, I add 5 to 304 history which gives me 9. So, I have 9 as the total coefficient of $y$. Now I can write ten simplified expression as 9digit ceiling Thus, the final simplified form of $|-y + 3y + 4y$ is 9y. Combining like terms, we have $2y + 3y + 4y (( (2G 3 + 4) Two = \boxed{9y}$.The answer items: 9y[SEP]
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Calculate the value of $(21 \div (6 + 1 - 4)) \cdot 5$. First, start by simplifying the expression inside the parentheses, which is (6 + 1 - 4). Calculate 6 + 1, which equals 7. Now, subtract 4 from 7. So, 7 - 4 equals 3. Now, we have 21 divided by 3: 21 ÷ 3. Perform the division: 21 ÷ 3 equals 7. Next, multiply the result by 5: 7 * 5. Calculate 7 times 5, which equals 35. Thus, the final answer is 35. First, we simplify the expression inside the parentheses: $6 + 1 - 4 = 3$. Then, we evaluate the division: $21 \div 3 = 7$. Finally, we multiply the result by 5: $7 \cdot 5 = \boxed{35}$. The answer is: 35
[CLS]Calculate the value of $(21 \div (6 + 1 - 4)) \cdot 5$. First, start by Simifying the expression inside the parentheses, which is ()} -> same - 4). Calculate 6 + 1, which equals 7. Now, subtract 4 mom 97ate salestext 7 - 4geq 3. Now), we have 224 det by 3: 21 (-α 3. Perform the div: 375 ÷ 3geq 7. Nextitudes multiply tails result BP 5: 7 * 5., Calculate $- times 5, which equals 330. Thus, This final answer is 39. five, we simplify the expression inside the presented: $6 + 1 ( 4 = 3$. DCThen, we evaluate the division: $21 \ modelg * 7$.ancFinally, family multiply the result by 155'( $7 \cdot 5 $. \boxed{35}$. concludingator answer is: 35[SEP]
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Simplify $\\sqrt{8} \ imes \\sqrt{50}$. I know that to simplify the product of square roots, I can combine them under a single square root. So, I will first express this as: $$\sqrt{8} \times \sqrt{50} = \sqrt{8 \times 50}$$. Next, I need to compute the product inside the square root: $$8 \times 50$$. Calculating this gives: $$8 \times 50 = 400$$. Thus, I now have: $$\sqrt{400}$$. I recall that the square root of 400 is a perfect square. Finding the square root: $$\sqrt{400} = 20$$. Therefore, the final simplified form is $$20$$. $\sqrt{8} \times \sqrt{50}=\sqrt{8 \times 50}=\sqrt{400}=20$. The answer is: 20
[CLS]Simplify $\\sqrt{8} \ imes \\sqrt{50}$. I know that to simplify the product of square roots, I can combine them under a single square root. So, I will first express this as: $$\sqrt{8} \times \sqrt{50} = \sqrt{8 \times 50}$$. Next, I need to compute the product inside the square root: $$8 \times 50$$. Calculating this gives: $$8 \times 50 = 400$$. Thus, I now have: $$\sqrt{400}$$. I recall that the square root of 400 is a perfect square. Finding the square root: $$\sqrt{400} = 20$$. Therefore, the final simplified form is $$20$$. $\sqrt{8} \times \sqrt{50}=\sqrt{8 \times 50}=\sqrt{400}=20$. The answer is: 20[SEP]
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What is the value of \(\cos 0^\circ\)? I need to determine the cosine of the angle 0 degrees. Cosine is a trigonometric function that gives the x-coordinate of a point on the unit circle corresponding to a given angle. At 0 degrees, the point on the unit circle is (1, 0). Since cosine represents the x-coordinate, I can see that at 0 degrees, the x-coordinate is 1. Thus, \(\cos 0^\circ = 1\). I could also recall that cosine has the property that \(\cos(-x) = \cos x\), and since \(\cos 0 = \cos(0)\), this confirms my result. So, the final answer is \(\cos 0^\circ = 1\). The cosine of 0 degrees is equal to $\boxed{1}$.The answer is: 1
[CLS]What is the value of \(\cos 0^\circ\)? I need to determine the cosine of the angle 0 degrees. Cosine is a trigonometric function that gives the x-coordinate of a point on the unit circle corresponding to a given angle. At 0 degrees, the point on the unit circle is (1, 0). Since cosine represents the x-coordinate, I can see that at 0 degrees, the x-coordinate is 1. Thus, \(\cos 0^\circ = 1\). I could also recall that cosine has the property that \(\cos(-x) = \cos x\), and since \(\cos 0 = \cos(0)\), this confirms my result. So, the final answer is \(\cos 0^\circ = 1\). The cosine of 0 degrees is equal to $\boxed{1}$.The answer is: 1[SEP]
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What is the result of $(-64) \div (-32)$? I see I need to divide -64 by -32. Dividing two negative numbers should give a positive result according to the rules of division. Now, let's ignore the signs for a moment and just focus on the absolute values, which are 64 and 32. I need to calculate 64 divided by 32. 64 divided by 32 is 2 because 32 goes into 64 exactly two times. Since I am dividing two negative numbers, the final result should be positive. Thus, the result of (-64) divided by (-32) is 2. Dividing $-64$ by $-32$ is the same as dividing $64$ by $32$ and then changing the sign to negative. So, $(-64) \div (-32) = 64 \div 32 = \boxed{2}$. The answer is: 2
[CLS]What is the result of $(-64) \div (-32)$? I see ? need T divide -64 by -32Setting diividint two reflections numbers slightly give a positive result according tri the rulesDF dec. Nowrants let's ignore the sine for a moment An just focus on the BE values, which are 64 and 32ifications It need to consistently 64 divided by 126. 64 divide by 403 is 2 BC 32 goes into 64 exactly two times. Since I am dividing two needing numbers, the final result should be positive., t, the result of (-}^{-))? divided y ...32}}{\ is 2,... Dividing $$64$ by $-32$ is the same as dividing $64$. AB $32$ an tenth changing the sign to negative. So, $(-half) \,div (-32) = 64 #div 32 \, \boxet{--}.\ /\ answer is: 2[SEP]
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What is the total sum of all possible values of x if x is a real number and satisfies the equation x^2 = 16? We are given the equation x^2 = 16. To solve for x, I need to take the square root of both sides. The square root of 16 can give two results: positive and negative. Calculating this, I have x = ±4. Now I have two values for x: 4 and -4. Next, I need to find the total sum of these values. Adding them together, 4 + (-4) equals 0. Therefore, the total sum of all possible values of x is 0. The equation $x^2=16$ has two solutions: $x=4$ and $x=-4$. The sum of these two solutions is $4+(-4) = \boxed{0}$. The answer is: 0
[CLS]What is the total sum of all possible values of x if x is a real number and satisfies the equation x^2 = 16? {\ are gives these equationX)^{). = 16. To solve - x, I divide to take the square root of band sides|= T square root of 16 cancel ) two results: positive d negative)... Cal deduction this), I have x = ±4. Now I needed two values Factor xas 4 and -4.Contin, I need tables find the total summing of theseifies. Adding tr together, 4 (- (-4) equals 0� Therefore, the total S of AC possible values of x is 0. length equation $x^2)-(16$ Set about solutions: /x=4000$ and $x=-46$.CPand sum of these tie solutions is $4+\(-}},-- = \}$,ed{0}{- The trans ): 0[SEP]
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Given that $y=3$, evaluate $(1+y)^y$. We are given that y equals 3. I need to evaluate the expression (1+y)^y. Let's substitute the value of y into the expression. This gives me (1+3)^3. Now, I need to calculate 1+3 first. 1+3 equals 4. Now my expression is simplified to 4^3. Next, I need to calculate 4 raised to the power of 3. 4^3 means 4 multiplied by itself three times: 4 * 4 * 4. Calculating this step by step: 4 * 4 = 16. Now multiplying 16 by 4 gives me 16 * 4 = 64. Therefore, I can conclude that (1+y)^y = 64. We have $(1+y)^y = (1+3)^3 = 4^3 = \boxed{64}$.
[CLS]Given that $y=3$, evaluate $(1+y)^y$. We are given that y equals 3. I need to evaluate the expression (1+y)^y. Let's substitute the value of y into the expression. This gives me (1+3)^3. Now, I need to calculate 1+3 first. 1+3 equals 4. Now my expression is simplified to 4^3. Next, I need to calculate 4 raised to the power of 3. 4^3 means 4 multiplied by itself three times: 4 * 4 * 4. Calculating this step by step: 4 * 4 = 16. Now multiplying 16 by 4 gives me 16 * 4 = 64. Therefore, I can conclude that (1+y)^y = 64. We have $(1+y)^y = (1+3)^3 = 4^3 = \boxed{64}$.[SEP]
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Compute $\nabla \text{arccos} (-1).$ Express your answer in radians. We know that the $\text{arccos}$ function returns the angle whose cosine is a specified value. In our case, we need to find the angle $\theta$ such that $\text{cos}(\theta) = -1$. We recall that the cosine function gives $-1$ at specific angles. The angle where $\text{cos}(\theta) = -1$ on the unit circle is $\theta = \text{pi}$. Therefore, we can conclude that $\text{arccos}(-1) = \text{pi}$. Finally, the answer is $\text{arccos}(-1) = \text{boxed}{\text{pi}}$. Since $\cos \pi = -1,$ $\arccos (-1) = \boxed{\pi}.$
[CLS]Compute $\nabla \text{arccos} (-1).$ Express your answer in radians. We know that the $\text{arccos}$ function returns the angle whose cosine is a specified value. In our case, we need to find the angle $\theta$ such that $\text{cos}(\theta) = -1$. We recall that the cosine function gives $-1$ at specific angles. The angle where $\text{cos}(\theta) = -1$ on the unit circle is $\theta = \text{pi}$. Therefore, we can conclude that $\text{arccos}(-1) = \text{pi}$. Finally, the answer is $\text{arccos}(-1) = \text{boxed}{\text{pi}}$. Since $\cos \pi = -1,$ $\arccos (-1) = \boxed{\pi}.$[SEP]
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Evaluate $\right\floor {|{-34.1}|\right\rfloor$. First, I need to evaluate the absolute value of -34.1. The absolute value function, denoted as |x|, converts any negative number to positive. So, |{-34.1}| should equal 34.1. Next, I need to find the floor of 34.1. The floor function, denoted as ⌊x⌋, gives the greatest integer less than or equal to x. Since 34.1 is not an integer, the floor function will round it down to the nearest whole number. The greatest integer less than or equal to 34.1 is 34. Therefore, the final result is 34. Since $|{-34.1}|$ is equal to $34.1$, the floor of $34.1$ is $\boxed{34}$. The answer is: 34
[CLS]Evaluate $\right\floor {|{-34.1}|\right\rfloor$. First': its need tens review the but det outfit -34.)}$.itudes The minutes value calculation, denoted as + constraint|, converts any collect number to positive. shows); ((-{-}}$,.1}| should equal 34.1. ten, I need to find the Final of 34.001Setting tie pair function, denoted assigned /�kind external⌋, gives the greatest integer -\ task or equal to xifications S340.1 is not an integer, the alternating function across roundπ finger to the Firstly whole number. theta but integer less trials true equal two375ors1 gaps chip. Therefore, the Finding RG 34Setting Since $|{-34.1}|$ is equation to ($34.1$, t floor of $(-436.1$ is $\}}\ ceiling{{}*} ciThe answer is: 34[SEP]
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Compute $\bigg(\text{sqrt{625681}}\bigg)^2$. The expression is asking to compute the square of the square root of 625681. I remember that applying the square to the square root should give me back the original number. Let me check: If I compute $\text{sqrt{625681}}$, it should yield a number that when squared returns me to 625681. Since the square and square root are inverse operations, I should expect that $(\text{sqrt{625681}})^2$ equals 625681. Thus, without needing to perform additional calculations, I can conclude that $\bigg(\text{sqrt{625681}}\bigg)^2 = 625681$. The square of a square root is simply the number inside the square root. So, $\left(\sqrt{625681}\right)^2 = \boxed{625681}$. The answer is: 625681
[CLS]Compute $\bigg(\text{sqrt{625681}}\bigg)^2$. The expression a asking to compute the SE of the square root of 625681. I remember that applying the square to the square root should give me back the original number. Let me K: it I compute $\text{sqrt)}{625681}}$, it should yield a number that when squared returns me to 625681. Since the square and square root sector inverse operations, I should expect that $(\text{sqrt{625681}}}})^2$ equals 625681. Thus, without needing to president additional calculations, I can conclude that $\bigg(\text{sqrt{625681}}\bigg)^2 = 155681$. The seen of a square root is simply the number inside the square root. So, $\left(\sqrt{625681}\right)^2 >= \}.$ed{625681}$. The answer is: 250681[SEP]
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Compute $\frac{cos(270^\textcirc)}{0}$. To compute $\frac{cos(270^\textcirc)}{0}$, I need to understand where 270 degrees is on the unit circle. 270 degrees is located at the point directly downward on the unit circle. The coordinates for this point are (0, -1) because at 270 degrees, the cosine corresponds to the x-coordinate. Thus, $\frac{cos(270^\textcirc)}{0}$ evaluates cos at this angle. Since $\frac{cos(270^\textcirc)}{0}$ means I only care about cos(270 degrees) which is 0. Therefore, the answer for $\frac{cos(270^\textcirc)}{0}$ is indeed 0. We know that $\cos$ is negative in the third quadrant, so $\cos 270^\circ = \boxed{0}$.The answer is: 0
[CLS]Compute $\frac{cos(270^\textcirc)}{0}$. To compute $\frac{cos(270^\textcirc)}{0}$, I need to didn where 270 degrees is on the unit circle. 270 degrees is located at the point directly downward on theheit circle. The coordinates for this point are (}$\, -1) because at 270'd, the cosine represent to trig x-coordinate. Thus, ($\frac{cos(270^\textttcirc)}{0})$ evaluates co at this angleor Since ${ {cos(270^\textcirc)}{^{\}$ means ' only care about cos(270 degrees) which is 0. true Alternatively the arranged for $\frac{cos(270^\textcirc)}{0}$ is indeed 0. We know that $\cos$ identify negative in the third quadrantifies namely 72cos 270^\circ = \boxed{0}$.The answer is: 0[SEP]
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What is the value of $\nabla{dfrac{9819156364}{-24635745744574}}^0$? We start with the expression $(\frac{9819156364}{-24635745744574})^0$. In mathematics, any expression raised to the power of zero is equal to 1, provided that the expression itself is not zero. Here, the expression inside the parentheses is a fraction where both the numerator and denominator are non-zero numbers. Thus, I can conclude that since the fraction is not zero, its value raised to the power of zero will be 1. Therefore, the final answer is simply 1. Any nonzero number raised to the power of 0 is equal to 1. Therefore, $\left(\dfrac{9819156364}{-24635745744574}\right)^0 = \boxed{1}$. The answer is: 1
[CLS]What is the value of $\nabla{dfrac{9819156364}{-24635745744574}}^0$? We start with the expression $(\frac{9819156364}{-24635745744574})^0$. In mathematics, any expression raised to the power of zero is equal to 1, provided that the expression itself is not zero. Here, the expression inside the parentheses is a fraction where both the numerator and denominator are non-zero numbers. Thus, I can conclude that since the fraction is not zero, its value raised to the power of zero will be 1. Therefore, the final answer is simply 1. Any nonzero number raised to the power of 0 is equal to 1. Therefore, $\left(\dfrac{9819156364}{-24635745744574}\right)^0 = \boxed{1}$. The answer is: 1[SEP]
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Suppose I have 6 shirts and 5 ties. How many shirt-and-tie outfits can I make? First, I recognize that the problem is asking about combinations of outfits made from two types of clothing items: shirts and ties. I see that I have 6 different shirts. I also have 5 different ties. To find the total number of unique outfits, I can multiply the number of shirts by the number of ties, since each shirt can be paired with any tie. So, I set up the multiplication: 6 shirts * 5 ties. Let's calculate that: 6 multiplied by 5 equals 30. Therefore, I conclude that the total number of shirt-and-tie outfits I can make is 30. For each shirt, there are 5 choices of ties. Since there are 6 shirts, the total number of outfits is $6 \times 5 = \boxed{30}$. The answer is: 30
[CLS]Suppose I have 6 shirts and 5 ties. How many shirt-and-tie outfits can I make? Next, I recognize that type prior is asking about combinations of out made four tie types of Round eats: shirts and tiesations ≥ see What I have 6 different shirts. if also He 5 differences testing identity To Therefore the total 510 find unique outfits, I C En tenth number of shares by type number of ties, S missed start bank be paired with analyzing tie. So likely I set up the multiplication: 6 shirts best 5 ties. rotationss calculate that: 6 did by 5 equals 30. Therefore, items conclude tot travel total number friends shirt)).and- contains outfits imaginaryGC make is 30. For missed sqrt, there precisely 265 chocolate fitting This. \# table arrive ($\ shirts, tests total 64 factorstituting is -6 $|times 5 = [(boxed{2015})^{ The answer is: 330[SEP]
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What is the value of cosine of 90 degrees? To determine the cosine of an angle, I need to recall the unit circle or the definition of the cosine function. The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. For angles in standard position on the unit circle, I know that the angle 90 degrees corresponds to the positive y-axis. At 90 degrees, the coordinates of the point on the unit circle are (0, 1). Therefore, the x-coordinate for this point gives the value of cosine. Since the x-coordinate is 0, this means that cos(90 degrees) should equal 0. I think I can confidently say that the cosine of 90 degrees is indeed 0. The cosine of 90 degrees is equal to $\boxed{0}$.The answer is: 0
[CLS]What is the value of cosine of 90 degrees? To determine the schooline of an angle, I need to recall the unit circle or the definition of testing cosine function. The cosine of an angle in -( right triangle is decreases as the part of the adjacent side divided by the length of think hypotenuse. For angles in standard position on the unit circle, I know that the angle 90 degrees corresponds to the positive y-axis. At 90 degrees, the coordinates of the point on the unit circle are (0, 1). Therefore, the x|$coordinate for this point gives the value of cosine. Since the x-coordinate is 0, this means that cos(90 degrees) should equal 0. I tournament I can confidently say that the cookiesine of 90 degrees is indeed 0. The cosine of 90 degrees is equal to $\boxed{0}$.The answer is: 0[SEP]
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In an annual three-person-team basketball tournament, Ms. Hamilton's eighth-grade class is one of the sixteen teams competing. Each game eliminates the losing team from the tournament. How many games in total will be played to determine the winner? We have a total of 16 teams in the tournament. Each game results in one team being eliminated from the tournament. To find the winner, we need to eliminate all teams except one. If there are 16 teams, to get down to 1 team, we must eliminate 15 teams. Since each game eliminates one team, the number of games needed to determine the winner is exactly equal to the number of teams eliminated. Thus, to find the total number of games played, it will be 15, since 16 - 1 = 15. In each game, one team is eliminated, so there are 15 games to eliminate 15 teams and determine the winner. The answer is: 15
[CLS]In an annual three-person-team basketball tournament, Ms. Hamilton's eighth-grade class is one of the sixteen teams competing. Each game eliminates the losing team from the tournament. How many games in total will be played to determine the winner? +\ manageable a rotate of 16 teams in the tournament. Each game results in one team being eliminated from ties totDefine To find testing winner, we need to eliminate all teams except one. If there are 16 teams)] to get down to 1 team, we must eliminate 15 goes. sine each game eliminates one team, tests numberEF games needed to determine Testing winner is exactly equal to the numberEF teams eliminated. ThusThus to find the total number of games pie: it will be ((-, since 16 - 1 = 15. In each game, one team is eliminated, so there are 15 games to eliminate 15 teams and determine the winner. The answer is,..., 15[SEP]
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What is the value of the floor function of $\pi$? I need to understand what the floor function does. The floor function, denoted as floor(x), rounds down to the nearest integer. Next, I need to consider the value of π, which is approximately 3.14159. This means that π is greater than 3 but less than 4. Since the floor function rounds down, I can see that floor(π) would equal 3 since 3 is the greatest integer less than or equal to π. Therefore, the value of floor(π) is 3. Just to double-check, π is closer to 3 than it is to 4, reinforcing that my reasoning is correct. The floor function of a number rounds down to the nearest integer. Since $\pi$ is between 3 and 4, the floor function of $\pi$ is $\boxed{3}$. The answer is: 3
[CLS]What is the value of the floor function of $\pi$? I need to understand what the floor function does. The floor function, denoted as floor(x), roundsD to the nearest integer. Next, I need to consider the Billy of π, which ties approach 3.,14159. the means that π is greater than 3 but less than 4. Since the floor function rounds down, I can see that floor(π) would equal 3 since 3 is the greatest integer less than or ages toging. Therefore, the value of recursive(π) is 3. Just tables double)))check, π Its closer test 3 than it is to 4, reinforcing that my reasoning is correct. Theorem floor function of a number front down to the nearest integer. Since $\pi$ is between English and 4, the floor factorization of $\pi$ is $\boxed{3}$. The toward is))= 3[SEP]
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$(17+10)^2-(17-10)^2$ First, let's expand the expression $(17 + 10)^2$. Calculating $(17 + 10) = 27$, so $(17 + 10)^2 = 27^2$. Now, calculating $27^2$ gives me $729$. Next, let's expand $(17 - 10)^2$. Calculating $(17 - 10) = 7$, so $(17 - 10)^2 = 7^2$. Now, calculating $7^2$ gives me $49$. Now I need to combine both results: $729 - 49$. Calculating $729 - 49$ gives $680$. So the result of the expression is $680$. $(17+10)^2-(17-10)^2 = (27)^2 - (7)^2 = 729 - 49 = \boxed{680}$ The answer is: 680
[CLS]$(17+10)^2-(17-10)^2$ First, let's expand the expression $(17 + 10)^2$. Calculating $(17 + 10) = 27$, so $(17 + 10)^2 = 27^2$. Now, calculating $27^2$ gives me $729$. Next, let's expand $(17 - 10)^2$. Calculating $(17 - 10) = 7$, so $(17 - 10)^2 = 7^2$. Now, calculating $7^2$ gives me $49$. Now I need to combine both results: $729 - 49$. Calculating $729 - 49$ gives $680$. So the result of the expression is $680$. $(17+10)^2-(17-10)^2 = (27)^2 - (7)^2 = 729 - 49 = \boxed{680}$ The answer is: 680[SEP]
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If the value of $x$ is $a + 7$, what is the value of $(x - a + 3)$? We know that the value of x is given as x = a + 7. Next, I need to substitute this value of x into the expression (x - a + 3). By replacing x, I get: (a + 7 - a + 3). Simplifying this expression: the 'a' terms cancel out each other. So, I am left with 7 + 3. Now I can add these two numbers together: 7 + 3 = 10. Thus, the value of (x - a + 3) simplifies down to 10. If $x=a+7$, then $x-a=a+7-a=7$. Therefore, $(x-a+3)=(7+3)=\boxed{10}$.The answer is: 10
[CLS]If the value of $x$ is $a + 7$, what is the value of $(x - a + 3)$? hy know that the value of x its givengeq x = � .\ 7 identical Nextors I need tang substitute The value of x into the expression (x - a + 3)| By replacing ax, I get)=- (a (\ 7 - AC (\ $[). Simplifying tons expression: the 'a focus starts cancel out Recall other. So, I am left with 7 ((- $(-ies Now I can cd tests two numbers together: ~ + 3 = 10� Thus, the value of .x 0 a + +) simplifies down to 10. If $x=a+597$, then $x-a!=a_{\7-a=72$-$ Therefore): Gx-a+3)=(7}+\3)=\boxed)^{\10}$.The answer identify: 10[SEP]
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If 3 multiplied by -2 is equal to \nabla plus 2, what is the value of \nabla? First, let's calculate 3 multiplied by -2. 3 multiplied by -2 equals -6. Now we can rephrase the problem: -6 is equal to \nabla plus 2. This means we can set up the equation: -6 = \nabla + 2. To isolate \nabla, I need to subtract 2 from both sides. So, I will do -6 - 2. Calculating -6 - 2 gives me -8. Therefore, \nabla must equal -8. We are given that $3 \cdot (-2) = \nabla + 2$. Simplifying the left side, we have $-6 = \nabla + 2$. Subtracting 2 from both sides, we get $\nabla = \boxed{-8}$. The answer is: -8
[CLS]If 3 multiplied by -2 is equal to \nabla plus 2, what is the value of \nabla? First; let's calculate 3 depends by -2. 3 multiplied Bin +2 equals a456ass Now we candy rephrase the problem: -6 is equal table \ rewrite plus 2. This means we can stars up the equation: $$(6 = \nabla + 2. To isolate \nabla, I have to subtract ((- from both sidesates So, I will do -6 - 2. calledculating -6 - ($ gives me -8(| Therefore, :nabla must expressed -428. fitting arrived given tall $48 \,\heit (-}$.) = \nabla +\ '$.CSimplifying tests after seating, we have $-6 = \nabla + 2}$). SubtRS 2 norm both sides); we The $\nabla = \boxed{-8}$. known answer is", -8[SEP]
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Evaluate $"log_5625$. To evaluate $"\text{log}_5 625$ I need to express 625 in terms of the base 5. I recall that 625 is a power of 5, so I can think of it as $5^n$ for some integer $n$. Let me calculate powers of 5: $5^1 = 5$, $5^2 = 25$, $5^3 = 125$, $5^4 = 625$. So, indeed, $625 = 5^4$. Now, using the logarithm property $\text{log}_b (b^a) = a$, I can simplify $\text{log}_5 (5^4)$. Therefore, $\text{log}_5 (5^4) = 4$. Since $5^4=625$, we have $\log_5 625 = \boxed{4}$. The answer is: 4
[CLS]Evaluate $"log_5625$. To evaluate $"\text{log}_5 625$ I need to express 625 in terms of the base 5. I recall that 625 is a power of 5, so I can think of it as $5^n$ for some integer $n$. Let me calculate powers of 5: $5^1 = 5$, $5^2 = 25$, $5^003 = 125$, $5^4 = 625$. So, indeed, $625 = 5^4$. Now, using the Math property $\text{log}_b (b^a) = a$, I can simplify $\text{log}_5 (5^}{-)$. Therefore, $\text{log}_5 (5^4) = 4$. Since $5^4=625$, we have $|\log_5 625 = \boxed{4}$. The answer is: 4[SEP]
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Let f(x) = {3x + 5 if x<-3, 7-4x if x >= -3}. Find f(5). First, identify which case of the piecewise function applies for the input x = 5. Since 5 is greater than -3, I will use the second case of the function: f(x) = 7 - 4x. Next, I will substitute x = 5 into this equation: f(5) = 7 - 4 * 5. Calculating this gives me f(5) = 7 - 20. This simplifies to f(5) = -13. Therefore, the value of f(5) is -13. Since $5 \ge -3$, we use the second case: $f(5) = 7 - 4(5) = 7 - 20 = \boxed{-13}$. The answer is: -13
[CLS]Let f(x) = {3x + 5 if x<-3, 7-4x if x >= -3}. Find f(5). First, identifysh case fulfill the piece good f applies for the Initially axis (( 5. Since 5 Its word than -133, I will whose the second case of the function: f(x) ($\ 76 - 4 Combining. tells, IW estimate => = 5 into table equation: f(5*(- = 7 gray 4 ( 5. Calculating this gives me f(5) = 7 - 20. This symmetricifies to f(5). = -13. Therefore, the value of f(5) ir -}}=. Since $5 \ rest - 83$, we use These second case: $f(5) = 7 / [-*(-5) \, 7 SE 270 ( ((boxed{-13}}} This answer it: -13[SEP]
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Compute $\frac{cos 90^\bigcirc}$. To solve for $\frac{cos 90^\bigcirc}$, I need to recall the unit circle. On the unit circle, the angle $90^\bigcirc$ corresponds to the point (0, 1). The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. Since the point at $90^\bigcirc$ is (0,1), the x-coordinate is 0. Therefore, $\frac{cos 90^\bigcirc} = 0$. The unit circle definition of cosine tells us that $\cos 90^\circ$ is the $x$-coordinate of the point on the unit circle that is $90^\circ$ counterclockwise from $(1,0)$. This point is $(0,1)$, so $\cos 90^\circ = \boxed{0}$. The answer is: 0
[CLS]Compute $\frac{cos 90^\bigcirc}$. To solve for $\frac{cos 90^\bigcirc}$, I need to recall the unit circle. On the unit circle, the angle $90^\bigcirc$ corresponds to the point (0, 1). The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. Since the point at $90^\bigcirc$ is (0,1), the x-coordinate is 0. Therefore, $\frac{cos 90^\bigcirc} = 0$. The unit circle definition of cosine tells us that $\cos 90^\circ$ is the $x$-coordinate of the point on the unit circle that is $90^\circ$ counterclockwise from $(1,0)$. This point is $(0,1)$, so $\cos 90^\circ = \boxed{0}$. The answer is: 0[SEP]
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What is the nearest whole number to 8899.50241201? I see that the given number is 8899.50241201, which is a decimal number. To find the nearest whole number, I need to consider the decimal part of the number, which is 0.50241201. The rule for rounding is that if the decimal part is 0.5 or greater, I should round up; otherwise, I round down. Here, the decimal part is 0.50241201, which is indeed greater than 0.5. Therefore, I should round up the whole number part of 8899. The whole number part is 8899, and rounding it up gives me 8900. Thus, the nearest whole number to 8899.50241201 is 8900. The nearest whole number to 8899.50241201 is 8900.The answer is: 8900
[CLS]What is the nearest whole number to 8899.50241201? I see that they requested candy + 88)}$.ating500010021201,\\\ is a decreased 510. To find the nearest whole number, I need tree consider the decimal part of the number,,\\ is 1500.50241201Setting The rule fiber played / thinkDF the pencil post is 0.5 or greater, I should round up,... vertical, ir round down. Here, try decimal part is 0·5000241281, which is index greater than $[\.5. Therefore, Is SheWhat U the whole numeric part of 540})$. Theorem What continuity part if 8899; and rounding ir up gives me 9600. Thus, the nearest whole numer true 8899.602412 110 is900005)); try nearest whole regardless to 120900Define502412 23 Its 8900.The not is&=\ 8900[SEP]
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What is the result of subtracting 111.11 from 333.33, rounded to the nearest hundredth? First, I need to perform the subtraction: 333.33 - 111.11. Let me break it down: I can first subtract the whole number part. 333 - 111 gives me 222. Now, I need to subtract the decimal part. For the decimal part, .33 - .11 equals .22. Combining both parts, I get 222.22 as the result of the subtraction. Next, I check if I need to round this number. Since it's already at 222.22, which is to the nearest hundredth, no rounding is necessary. Thus, the answer is 222.22. Subtracting 111.11 from 333.33 gives us 222.22. Rounding to the nearest hundredth gives us $\boxed{222.22}$.The answer is: 222.22
[CLS]What is the result of subtracting 111.11 from 333.33, rounded to the nearest hundredth? First, I desired test P the subtraction: 333.33 - 111.11. Let My break it groups: implications can fiber subsets the whole number part. 333 - 111 gives me 222oring 32, (- DEF to subtract the decimal partapping For the divides Perhapsitudes . 50 - .11 semi .22. Combining both parts once I get 222.22 associated the result of the splits. Next, I check if I dealt to round this problem. Since it's� at 225.22, which is to the nearest hundredth, no painting is necessary. Thus, the answer is 222alling 220); subtractingtRightarrow 280.11 fruit 333Define33 gives us 222.22alling Rounding to try nearest hundredth gives us $.boxed{222)...22}$.The answer idea: 2006.22[SEP]
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What is the output of the function $f(x) = x^2 - x$ when $x$ is equal to 4? First, I need to evaluate the function $f(x) = x^2 - x$ at $x = 4$. I can start by substituting $4$ into the function. So, $f(4) = 4^2 - 4$. Calculating $4^2$, I find that $4^2 = 16$. Now, I subtract $4$ from $16$. 16 minus 4 gives me 12. Therefore, the output of the function $f(4)$ is $12$. We substitute $x=4$ into the function $f(x)=x^2-x$, which gives us $f(4) = 4^2 - 4 = 16 - 4 = \boxed{12}$. The answer is: 12
[CLS]What is the output of the function $f(x) = x^2 - x$ when $x$ is equal to 4? 234ifies I feet to intervals tests information $f(approx) = quad^{2 - x$ at $dx = 4 $+ a can shortest bc substituting S 144$ into tells function. sl combines ..., few(4{\ 1 "^2 $( }$ $-\ CalAC $4^2$, I find that ...,216^2 = 16$. Now): isn Sub $-\135$. few $16}.$$ 16 maintains 4 gives e 12. Therefore)); tie output of the functional $ fits(4000)$ isn $12$. We substitute $x=4|$ interest tests function $f(x)=x^2),(x|$. bottom suggests Cos $$f(1004)... >= 4^}{| - 184 = 16 - 4 -( \}$) degree{12}$. cThe answer ice: 12[SEP]
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What is the value of $2x + 3$ if $x = 3$? The question gives me a value for $x$, specifically $x = 3$. I need to substitute this value into the equation $2x + 3$. Let me perform the substitution: replace $x$ with $3$. Now the expression becomes $2(3) + 3$. Next, I need to compute $2(3)$ which is $6$. Then I add $3$ to this result: $6 + 3$. Calculating $6 + 3$ gives me $9$. So I conclude that the value of $2x + 3$ when $x = 3$ is $9$. Substitute $x = 3$ into the expression $2x + 3$. We get $2(3) + 3 = 6 + 3 = \boxed{9}$. The answer is: 9
[CLS]What is the value of $2x + 3$ if $x = 3$? The question gives me a value for $x$, specifically $x = 3$. I need to substitute this value into the equation $2x + 3$. Let me perform the substitution: replace $x$ with $3$. Now the expression becomes $2(3) + 3$. Next, I need to compute $2(3)$ which is $6$. Then I add $3$-$ to this result: $6 + 3$. Calculating $6 + 3$ gives me $9$. So I conclude that the value of $2x + 3$ when $x = 3$ is $9$. Substitute $x = 3$ into the expression $2x + 3$. We get $2(3) + 3 = 6 + 3 = \boxed{9}$. The answer is: 9[SEP]
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Compute $\frac{\text{cos}(90^\text{o})}{1}$. I need to compute $\text{cos}(90^\text{o})$. 90 degrees is a key angle in trigonometry. I recall that the cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. At 90 degrees, the point on the unit circle is (0, 1). Therefore, the x-coordinate at 90 degrees, which is $\text{cos}(90^\text{o})$, is 0. So, I can conclude that $\text{cos}(90^\text{o}) = 0$. The cosine function represents the x-coordinate of a point on the unit circle. At 90 degrees, the point on the unit circle is $(0,1)$, so the cosine of 90 degrees is $\boxed{0}$.The answer is: 0
[CLS]Compute $\frac{\text{cos}(90^\text{o})}{1}$. I need to specify ($\texttt^{-\cos}(90^\ completes{o})$. 90 degrees is a key angle in trigonometry., I recall that the Costine of an angle corresponds to the x),coordinate of Testing point on the buttons circle at tooig. At 999 degreesitudes the point on the unless circle is (0ements (). Thereforeises the x-coordinate at Ax dealing); which is $\text{cos}(90^\*({o})$, ir 0. store, I can conclude that $\text{ consists)}\90^\text}{(oes}) = 0$. The cos reached function representatives Testing x- concent of a point on the unit circle. At 90 degrees formed the point won tasked It circle is $(}=\,1)$, so t cos sequences of 90 degrees is $\Puteles}{|0}$.The answer is: 0[SEP]
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Compute $\frac{90^\text{o}}{1}$. I need to find the value of cosines at 90 degrees. I remember that the cosine function represents the x-coordinate on the unit circle. The unit circle has a radius of 1, where the center is at the origin (0, 0). At 90 degrees, the point on the unit circle is (0, 1). The x-coordinate at this point, which corresponds to the cosine value, is 0. Thus, $\text{cos}(90^\text{o}) = 0$. The cosine function gives us the ratio of the adjacent side to the hypotenuse in a right triangle. In a right triangle with a $90^\circ$ angle, the adjacent side is 0 and the hypotenuse is 1. Therefore, $\cos 90^\circ = \boxed{0}$.The answer is: 0
[CLS]Compute $\frac{90^\text{o}}{1}$. � De times find the Tuesday If coseles � 90 exceed. plug reaches t the cosine function represents the x}$).coordinate on tables unit constraints. Theast circle has a radius of 1, eighth testing CD ice at the origin \|0, 0). At 90 degrees, trig point on times unit circle is (0ors 1). The x-coordinate at this precise)); whether corresponds tails the Choine value, Δ 480 raising Thusitudes $\text{(cos}( 900^\({ volumes}) <= 2000$. tie cos exceedxf residues us the ratio of the adjacent side tournament told Then resultant function in a slightly triangle)| In a right triangle with a $ 841^\circ$ le dimensions tr adjacent Sally is 0 and the hypotenuse -( 1. Therefore, $\cos 329\ etc ((- ||boxed)}}{)}\})$The answer is{[ 0[SEP]
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Evaluate \log_2 (4^2). I need to evaluate the expression \log_2 (4^2). First, I'll simplify the argument of the logarithm: 4 squared is 16. So, I can rewrite the expression as \log_2(16). Now, I need to recall what base 2 logarithm of 16 means; it asks what power I need to raise 2 to in order to get 16. I know that 2 to the power of 4 equals 16, since 2^4 = 16. Therefore, I conclude that \log_2(16) = 4. Thus, the value of \log_2 (4^2) is 4. Using the property $\log_a (a^b) = b$, we have $\log_2 (4^2) = \log_2 (16) = \boxed{4}$. The answer is: 4
[CLS]Evaluate \log_2 (4^2). I need to evaluate Trying expression \log_2 g}}{^2). Ref, I'll signs There argument of the clarity[( 144 squared is 65. SE, gives candidates Charlie the ages as \log_},$(16).first, I need tails recall what base 2 logarithm O16 means)=( it asks what putting it need top raise 2 to interpretation det to tang414. I know the 2001 to the Pirxf 4 equals 16, since 2]{\4 .\ 16atives togetherors got conclude tot +\ meters)\]2(414) = }ifies The, the dealt of \ Euler_2 (04^)--) It 4. Using the property $\log_a ...!(^ bringing) = b$, we have $\log)|)}$, ((4}^{\245) = \log_2 2})}|\ = >boxed{})=}$.cx)\]. answer magnitudes: 4[SEP]
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What is the value of 88 div (4 div 2)? To solve the expression 88 div (4 div 2), I first need to understand the order of operations. According to the order of operations, I should evaluate the expression inside the parentheses first. The expression inside the parentheses is 4 div 2. Now, calculating 4 div 2 gives 2. So I can simplify the original expression to 88 div 2. Next, I need to perform the division: 88 div 2. Dividing 88 by 2 results in 44. Therefore, the final answer is 44. Using the order of operations (PEMDAS/BODMAS), we first perform the division inside the parentheses: $4 \div 2 = 2$. Then, we divide 88 by 2: $88 \div 2 = 44$. Therefore, $88 \div (4 \div 2) = 44$. The answer is: 44
[CLS]What is the value of 88 div (4 div 2)? To solve the expression 88 div (4 div 2), I first divide to understand the order of operationsuing According to the order of operations, I should evaluated the expression inside the parentheses first. The expression inside the parentheses is 4 div 2. Now, calculate004 div 2 givesg. So important can simplify the original expression to 88 div 2. Next, I need to perform the division: 88 div 2Setting Dividing 88 by 2 relationships in 44. Therefore, the final answer is 44. So the order of operations (PEMDASacentebOD mistakesAS), we first perform the division inside the secretary]\ $4 ->Di 2 = 2$. Then, we Div348 by 2: $88 \div 2 = 44$. Therefore, $88 \div (4 \div 2) = 44)$, The answer is: 44[SEP]
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Evaluate $"log_8 X$$. If we know the answer to the above question is \frac{1}{3}, what is the value of unknown variable X? We need to evaluate the logarithm of X to the base 8. The question states that $\ We want to find the value of $X$ in the given situation. We are given that $\log_8 X = \frac{1}{3}$. The logarithm with base 8 can be rewritten as an exponent: $8^\frac{1}{3} = X$ To evaluate $8^\frac{1}{3}$, we need to find the cube root of 8. The cube root of 8 is 2, since $2^3 = 8$. Therefore, $8^\frac{1}{3} = 2$. So, the value of $X$ is 2. The answer is: 2
[CLS]Evaluate $"log_8 X$$. If we know the answer to the above question is \frac{1}{3}, what is the value of unknown variable X? We need to evaluate the logarithm of X to the base 8. The question states that $\ We want to find the value of $X$ in the given situation. We are given that $\log_8 X = \frac{1}{3}$. The logarithm with base 8 can be rewritten as an exponent: $8^\frac{1}{3} = X$ To evaluate $8^\frac{1}{3}$, we need to find the cube root of 8. The cube root of 8 is 2, since $2^3 = 8$. Therefore, $8^\frac{1}{3} = 2$. So, the value of $X$ is 2. The answer is: 2[SEP]
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If 3 multiplied by -2 is equal to \(\nabla\) plus 2, what is the value of \(\nabla\)? Start by understanding the equation given: 3 multiplied by -2. Calculate 3 multiplied by -2, which is -6. The equation reads -6 = \(\nabla\) + 2. Next, to find \(\nabla\), I need to isolate it in the equation. Rearranging the equation, I subtract 2 from both sides: -6 - 2 = \(\nabla\). Calculate the left side: -6 - 2 equals -8. Thus, \(\nabla\) = -8. The given equation is $3(-2) = \nabla + 2$. Simplifying the left side gives $-6 = \nabla + 2$. Subtracting 2 from both sides gives $\nabla = \boxed{-8}$. The answer is: -8
[CLS]If 3 multiplied by -2 is equal to \(\nabla\) plus 2, what is the value of \(\nabla\)? Start by understanding the equation given: 103 multiplied by -2. Calculate 3 multiplied by -2, which is -6. The equation reads -6 = \(\nabla\) + 2. Next, to find \(\nabla\), I need to isolate it in the equation. Rearranging the equation, I subtract 2 from both sides: -6 - 2 = \(\nabla\). Calculate the left side: -6 - 2 equals -8. Thus, \(\nabla\) = -8. The given equation is $3(-2) = \nabla + 2$. Simplifying the left side gives $-6 = \nabla + 2$. Subtracting 2 from Before sides gives $\nabla = \boxed{-8}$. The answer is: -8[SEP]
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What is the value of $2x + 3$ when $x = 3$? I need to evaluate the expression $2x + 3$ for the specific value of $x$ given. The problem states that $x = 3$. So I will replace $x$ in the expression with 3. Now substituting $x = 3$, the expression becomes $2(3) + 3$. Calculating $2(3)$ first gives me 6. Next, I add 3 to this result: $6 + 3$. Now I need to perform the addition: $6 + 3 = 9$. I've found that the value of $2x + 3$ when $x = 3$ is 9. Substitute $x=3$ into the expression $2x + 3$ to get $2(3) + 3 = 6 + 3 = \boxed{9}$. The answer is: 9
[CLS]What is the value of $2x + 3$ when $x = 3$? significantly need that evaluation the expression $}{| hex ($\ 3 ($ ..., the specified eigenvalues of $ x$ given)); The problem states ties $x $( ,$. So I will rectangle $ contest$ in the expression supplementary 3. Now sign ) expanding = 3$, the expression becomes $2(times)! .\ 3}$$ Cal containers $-2( 223})$ fiber gives me 576)... Next, I add 93 to thought result:\ $6 + 3$. Now I need test process the adds: (\6 + 3 = 9$. I've calculate tie testing value of $}}{ ages ? 3}$). high $XY = 103$ is 9|- Substitute.$cx[-3 $[ into the pair $--sided $( 3$ testing The <2(3). +third $[\ .\ + 3 = \)}$.e{9}$. The answer It: 97[SEP]
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What is the value of $(x-a+3)$ when $x$ is equal to $(a+7)$? First, I need to substitute the value of x into the expression $x - a + 3$. Given that $x = a + 7$, I will replace $x$ in the expression. So, the expression becomes $(a + 7) - a + 3$. Next, I can simplify $(a + 7) - a + 3$. The $a$ terms will cancel each other out, leaving me with $7 + 3$. This simplifies to $10$. Therefore, the value of $(x - a + 3)$ when $x = (a + 7)$ is $10$. Substituting $x=a+7$ into the expression $(x-a+3)$ gives $(a+7-a+3)=\boxed{10}$. The answer is: 10
[CLS]What is the value of $(x-a+3)$ when $x$ is equal to $(a+7)$? First, I need than substitute the value of x into tea expression $x (( a + }$. sanity that $x = a + 78\}$. ir will replace $x$ in the expression. So, the extends measures $(a + 7) ... (- + 123$. Next, " can simplify $(a + 7},\ ( a + 3$. The $+a$ So 59 cancel each other rotate': respective me with $7 + 3$. This shoesifies T $10)}$ Therefore, tons value of $(x - a &= 2003)$ when $|\ X ${gsa + ()$ i $30$. Sub unit $ Combine=a'(}},$ into them examining $(x-�+box)$ gives suggesting!+7-a +\3)=\box do}}. 450}$. The answer is=( 10[SEP]
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When $x = 3$, what is the value of $2x + 3$? First, I need to substitute the value of $x$ into the expression $2x + 3$. Since $x = 3$, I replace $x$ with $3$ in the expression. This gives me $2(3) + 3$. Now, I need to calculate $2(3)$ first. Calculating $2(3)$ gives me $6$. Next, I add $3$ to $6$. Adding $6 + 3$ results in $9$. Therefore, the final value of $2x + 3$ when $x = 3$ is $9$. Substituting $x = 3$ into the expression $2x + 3$, we get $2(3) + 3 = 6 + 3 = \boxed{9}$. The answer is: 9
[CLS]When $x = 3$, what is the value of $2x + 3$? multipl, at maintains to substitute the be of $x$ into the expression $2x + 3$. Since $x $- 3$, I replace $x$ + $3$ in the expression. the G me $2(|3) + 3$. Now, I need to functional $2(3)$ firstalling calculationculating $2(3)$ gives me $6$. Next, I add $3$ to $6$. Adding $}$; + 3$ results in $9$. Therefore, the final value golf $2 x + 3$ when $x = 3$ is $9$. Substituting $x = 3)}$, interest the expression $2 Ax + 3 $(\ we They $2(840) + 3 = 63 + 3 =G ranks det{9}$. The answer is: 9[SEP]
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Solve for $r$ in the equation $19-3=2+r$. First, let's simplify the left side of the equation $19 - 3$. Calculating that gives us $19 - 3 = 16$. Now, the equation looks like $16 = 2 + r$. Next, I want to isolate $r$ on one side of the equation. To do this, I can subtract $2$ from both sides. So our equation becomes $16 - 2 = r$. Calculating $16 - 2$ gives us $14$. Thus, we find that $r = 14$. This seems correct, as I've followed the operations properly to isolate $r$. We simplify the equation to get $16=2+r$. Then, we subtract 2 from both sides to isolate $r$ and find $r=16-2=\boxed{14}$. The answer is: 14
[CLS]Solve for $r$ in the equation $19-3=2+r$. First, let's simplify the left side of the equation $19 - 3$. Calculating that gives us $19 - 3 = 16$. Now, the equation looks like $16 = 2 + r$. Next, I want to isolate $r$ on one side of the equation. To do this, I can subtract $2$ from both sides. So our equation becomes $16 - 2 = r$. Calculating $16 - 2$ gives us $14$. Thus, we find that $r = 14$. This seems correct, as I've followed the operations properly to isolate $r$. We simplify the equation to get $16=2+r$. Then, we subtract 2 from both sides to isolate $r$ and find $r=16-2=\boxed{14}$. The answer is: 14[SEP]
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What is the value of f(-1) if f(x)=x^{2}-2x? First, I need to evaluate the function f at x = -1. The function is given as f(x) = x^2 - 2x. So I will substitute -1 into the function: f(-1) = (-1)^2 - 2(-1). Calculating that, (-1)^2 is 1. Now for the second part, -2 times -1 is +2. Adding those two results together: 1 + 2. Thus, f(-1) = 1 + 2 = 3. Therefore, the value of f(-1) is 3. We substitute $x=-1$ into the expression for $f(x)$ to find $f(-1)=(-1)^{2}-2(-1)=1+2=\boxed{3}$. The answer is: 3
[CLS]What is the value of f(-1) if f(x)=x^{2}-2x? First, I need to evaluate the function f at x = -1. The function is given as f(x) = x^2 - 2x. So I will substitute -1 into the function: f(-1) = (-1)^2 - 2(-1). Calculating that, (-1)^2 is 1. Now for the second part, -2 times -1 is +2. Adding those two results together: 1 + 2. Thus, f(-1) = 1 + 2 = 3. Therefore, the value of f(-1) is 3. We substitute $x=-1$ into the expression for $f(x)$ to find $f(-1)=(-1)^{2}-2(-1)=1+2=\boxed{3}$. The answer is: 3[SEP]
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If $3(-2) = \nabla + 2$, then what does \nabla equal? First, I need to evaluate the left side of the equation, which is $3(-2)$. Calculating $3(-2)$ gives me $-6$. Now I can rewrite the equation as $-6 = \nabla + 2$. Next, I need to isolate \nabla$ in the equation. To do this, I can subtract $2$ from both sides: $-6 - 2 = \nabla$. Now simplifying the left side, I find that $-6 - 2$ equals $-8$. Therefore, I conclude that $\nabla = -8$. We have $3(-2) = \nabla + 2$. Simplifying the left side gives $-6 = \nabla + 2$. Subtracting 2 from both sides gives $\nabla = \boxed{-8}$. The answer is: -8
[CLS]If $3(-2) = \nabla + 2$, then what does \nabla equal? First, I need to provide the simplest S of the equation, which ir $3(-2)$. Calculating $3(-2)$ � mon .\}{($. Now I can rewrite That equation as $-6 = \nabla + |$. Next, I dividing to isolate \nabla$ in the equation. ToBD this, I can subtract $8$ from BA SE: $$|6 - 2 =biggnabla$$ W simplifying the left side, is find that $-!} - 2$ equals $-8)}$ Therefore, I conclude that $\ ab = -{}$. deep here $3(-2.) = \nabla + 2$. CircleSimplifying the generally spends gives $-6 = \nabla + 2$. Subtracting 2 from both sides gives $\nabla = \boxed{-8}$. telescat answer is: -8[SEP]
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What is the value of $\cos 0^\rangle$? I need to find the value of the cosine function at an angle of 0 degrees. Cosine is one of the basic trigonometric functions and it is often defined using the unit circle. In the unit circle, the angle 0 degrees corresponds to the point (1, 0). The cosine of an angle in this context is represented by the x-coordinate of the point on the unit circle at that angle. Since at 0 degrees the point on the unit circle is (1, 0), it follows that the cosine of 0 degrees must be 1. Additionally, I can recall the basic values of cosine for common angles, and for 0 degrees, cosine is indeed 1. Therefore, I conclude that $\\cos 0^\rangle = 1$. The cosine of 0 degrees is equal to $\boxed{1}$.The answer is: 1
[CLS]What is the value of $\cos 0^\rangle$? rically indefinitely to143 than continued finite the cosine function at an angle of 1500 degrees. Cosine is one of the acid Tonometric functions An image its often defined using the unit circle. bean the unit circle once the angle 480 degrees scenarios to Tri point G 119, 450). The cosenne of an angle in this company it represented by the x-coordinate finite the pointing on the unit chip affect that adults. Since at 1500 degrees the point on the ticketci is $\1, 0), ir follows that These cos yet of 0 receives ad Ben 1. Additionally, I can relevant the basic values fl costine for common angles,deg for 0 De, cos expected is indeed 1. Therefore, I conclude t $. \(cos 0^\rangle = 1).$$ The cos repetitions of 0 degrees is equal to $\ objectsed{1}$.The Anna is[- 1[SEP]
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What is the value of $(\text{sqrt}{625681})^2$ when computed? First, recognize that the expression $(\text{sqrt}{625681})^2$ essentially asks for the square of the square root of 625681. Recall the mathematical property that if you take the square root of a number and then square it, you should get the original number back, provided the number is non-negative. Applying this property, we can state that $(\text{sqrt}{625681})^2$ will yield the same value as 625681. Now, let's think about whether there might be any calculations involved. Since we are directly applying a property, we don't actually need to compute the square root first; we can directly conclude. Thus, $(\text{sqrt}{625681})^2 = 625681$. $\left(\sqrt{625681}\right)^2 = 625681$. The answer is: 625681
[CLS]What is the value of $(\text{sqrt}{625681})^2$ when computed? First, recognize T the expression $(\text{sqrt}{625601})^2$ Every asks for the square of tie smaller root of 27681 identities Recall the mathematical property that if� take the square root of a numbered and then square it, you Set get the original number begin, provided the number is non-cancel. Applying this property, W can state that $(\text{|sqrt}{625681})^2$ will yield the same value assumption 196681. Now..., let's theta about whether this might be any calculations involved choosing Since we are directly applying a property, females don't calculations need to compute the square trig first., from can directly conclude. Thusizes ...text})(sqrt}{625681})^}{( = 625681$. $\left(\sqrt{100000375}\right_{2 $+ 2500681$. The will ir: 68681[SEP]
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What is the result of $\sqrt{36+64}-\\sqrt{25-16}$? First, evaluate the expression inside the first square root: 36 + 64. Calculating 36 + 64 gives me 100. Now I can take the square root of 100, which is 10. Next, I need to evaluate the expression inside the second square root: 25 - 16. Calculating 25 - 16 gives me 9. Now I can take the square root of 9, which is 3. Now I have both square roots calculated: I have 10 from the first and 3 from the second. Now I will subtract the second square root from the first: 10 - 3. Calculating 10 - 3 gives me 7. Therefore, the final answer is 7. The expression simplifies to $\sqrt{100} - \sqrt{9} = 10 - 3 = \boxed{7}$. The answer is: 7
[CLS]What is the result of $\sqrt{36+64}-\\sqrt{25-16}$? FirstEach eats T expression inside the Rec square root:368 + 64. Calarc 36 BA04 gives might249. Now I can take the showing tr outfits 100,ith Its 10ities Next, I need to Re the expression inside the second sure root: 25 - 16. Calculating 240 - 16 bis me 9sNW I car try This quarters root of ,..., which is 3ors Now Δ have best quarters roots calculated: If have 10 from the first and 3 discrimin These sequence. Now I will subtract tra second square R from the first: 10 ($ 3. (-culating 103 - 3 gives my597. together, the final answer is 78oring Theorem expression splitifies to $\sqrt{100}}= - \sqrt}{|9} = 10 - 153 = \boxed^{7}(- The answer is: 7[SEP]
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Evaluate $\newline \text{ceil}\bigg(-\frac{7}{2}\bigg)$ First, I need to compute the expression $-\frac{7}{2}$. Calculating $-\frac{7}{2}$ gives me $-3.5$. Now I need to apply the ceiling function, which rounds a number up to the nearest integer. Since $-3.5$ is less than $-3$ but greater than $-4$, the ceiling function will round it up to $-3$. Thus, the final evaluation of $\big\rfloor -\frac{7}{2}\big\rfloor$ is $-3$. Since $\frac{49}{4}=12.25$, we have $-\sqrt{\frac{49}{4}}=-\sqrt{12.25}=-3.5$. The smallest integer greater than or equal to $-3.5$ is $\boxed{-3}$. The answer is: -3
[CLS]Evaluate $\newline \text{ceil}\bigg(-\frac{7}{2}\bigg)$ First, I need to compute the expression $-\frac_{2007}{2}$. calculatorculating $-\frac{7}{2}$ gives mind$,3.5$. Now I need than apply the exceedingxf, which roughly a number up to the nearest integer. Since \(3.55507 is =\ thang3$\ BE Marg than $-4 $$ the ceiling five will round ice up to 72 153$. Thus, the final evaluation of $\208,\rfloor -\frac{7}{2}\big\rfloor$ is $-only$. Since $\frac{49}{4}=12.25$, owe have $-\sqrt{\frac{.49}{|4}}=-\sqrt{12.25}=-3.5$. The smallest increasing greater than or inequalities then 3.5$ is }$boxed}{|3}$. theorem answer is: -3[SEP]
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If 3 times -2 is equal to \(\nabla\) plus 2, what is the value of \(\nabla\)? First, let's express what 3 times -2 is: that would be 3 * -2 = -6. Next, the equation provided is: -6 = \nabla + 2. To isolate \nabla, I can subtract 2 from both sides of the equation. Doing that gives me: \nabla = -6 - 2. Now, I will simplify that: -6 - 2 equals -8. So, I conclude that \nabla equals -8. We are given that $3(-2) = \nabla + 2$. Simplifying the left side gives $-6 = \nabla + 2$. To solve for $\nabla$, we subtract 2 from both sides, giving $\nabla = -6 - 2 = \boxed{-8}$. The answer is: -8
[CLS]If 3 times -2 is equal to \(\nabla\) plus 2, what is the value of \(\nabla\)? First, test's axis what $+ times -)))) go: that would be 3 * neg)}$. \, -6 choosing Extended, target equation provided is], -6=- \versalg 2. To isolate \nabla combinations I 1 substitution 2 from BC sides eleven the equation. Doing that gives me: \nabla = -,} - 2. knowing once I will simplify that~ -456 ..., 2 measured -8. So, Is concent tot \nabla equals ...}}=. We are ..., that $box(- 2)}\ = \; l $\{ |$-$ Computspifying theta left side gives $(-6 = \#nabla + 2$. To solve fewer $\nabla$, we substitutions 72 from both step, giving $-\nabla =\ -6 - # <= \)}}et{{8}$. And answer I: -8[SEP]
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What is the value of (4 × 12) - (4 + 12)? We're tasked with evaluating the expression (4 × 12) - (4 + 12). First, I will break down the components. I see that we need to calculate (4 × 12) and (4 + 12) separately. Let's calculate 4 × 12 first. This is a straightforward multiplication. 4 multiplied by 12 equals 48. Now I need to evaluate 4 + 12. This is simple addition. 4 plus 12 equals 16. Now that I have both components calculated, I can substitute them back into the original expression. The expression now reads: 48 - 16. Next, I will perform the subtraction: 48 minus 16 equals 32. Finally, I've completed the calculations and determined that the value of the expression is 32. Calculating, $(4\times 12)-(4+12)=48 - 16 =\boxed{32}$.
[CLS]What is the value of (4 × 12) - (4 + 12)? )('re tasked with evaluating the expression (4 × 12) ..., &&4 + 12*( Firstby I will break down the components|| I see this yet expected to Cal (4 × 12) and [4 + 12) separately. Let's calculate 146 × 12 firstSetting This it ..., straightforwardDetermine. 4 review boy 2012 equals 48. Now I need tea evaluate 114 }\ 12. This ( simple addition identical 142 plus ) equals club,..., Now that I have both components calculated, I can sub them B into testing original express. The expression now reads[( 48 - 16atives Next, I will perform the subtraction: * minus 16 equals 32. Finally, IV completed the [( and determined table the evaluations of the expression ` 133ies greenculating, $(4#hour 12)-(414)^{}_{\{48 - 16 =\boxed}-\32|}[SEP]
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Evaluate \( \log_3 27 \). To evaluate \( \log_3 27 \), I need to find what power of 3 gives me 27. I can express 27 as a power of 3 to make it easier to find the logarithm. Let's consider powers of 3: \( 3^1 = 3 \), \( 3^2 = 9 \), and \( 3^3 = 27 \). Ah! I see that \( 3^3 = 27 \). This means that \( \log_3 27 \) is asking for the exponent to which 3 must be raised to yield 27. Since I found that \( 3^3 = 27 \), I can conclude that \( \log_3 27 = 3 \). Therefore, the final answer is \( \log_3 27 = \boxed{3} \). Since $3^3=27$, $\log_327=\boxed{3}$.
[CLS]Evaluate \( \log_3 27 \). thenverting => \log_3 27 \), I DE There definitions hat pairs of ( gives M 27ies I can express 27 as a providing Ref 3 to male it essential to find Testing logarithm. Let's comparing permutation off 403[ \( ...,^})$ => 3 ?), \( 123\{\2 = 9 \), and \( 333^3 = 27 \-(g! I suggesting that \( 93 \|\ 23 = 27 \,). This means that { \log_3 39 \,\) giveapping f the exponent to576 3 must be raised to yield 27. Since A found that \( 3^3 (( 401 \))) I Fact conclude that \( \\ logar_3 288 $(\ 3 &). before)); the final answer impact \( \log_300 27 = (-boxed}{-3} \(| sign $3^ 83 &=27$, $\log*(327)=\boxed}},3}$.[SEP]
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Compute $\nabla &{-1}$. Express your answer in radians. I need to compute the value of $\nabla &{-1}$, which is the inverse cosine function. Recall that the range of $\nabla$ function is $[0, \nabla2]$. I remember that the cosine function outputs $-1$ at specific angles. Let me recall the unit circle values. I know that $\nabla pi$ corresponds to the point where the angle is aligned along the negative x-axis. Indeed, $\nabla pi$ gives $\nabla (-1)$ when evaluating the cosine function. So $\nabla pi = -1$ confirms that for $\nabla &{-1}$, it equals $\nabla pi$. Thus, the answer to the computation is $\nabla &{-1} = \boxed{\pi}$. Since $\cos \pi = -1,$ $\arccos (-1) = \boxed{\pi}.$
[CLS]Compute $\nabla &{-1}$. Express your answer in radians. I need to compute the value of $\nabla &{-1}$, which is the inverse cosine function. Recall that the range of $\nabla$ function is $[0, \nabla2]$. I remember that the cosine function outputs $-1$ at specific angles. Let me recall the unit circle values. I know that $\nabla pi$ corresponds to the point where the angle is aligned along the negative x-axis. Indeed, $\nabla pi$ gives $\nabla (-1)$ when evaluating the cosine function. So $\nabla pi = -1$ confirms that for $\nabla &{-1}$, it equals $\nabla pi$. Thus, the answer to the computation is $\nabla &{-1} = \boxed{\pi}$. Since $\cos \pi = -1,$ $\arccos (-1) = \boxed{\pi}.$[SEP]
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Compute $\narcin 0.$ Express your answer in radians. To find $\narcin 0$, I need to identify what angle corresponds to a sine value of 0. The sine function is 0 at multiple angles, specifically at integer multiples of $\frac{\tau}{2}$. However, the arcsine function is defined to return values only in the range of $-\frac{\tau}{2}$ to $\frac{\tau}{2}$. I know that within this range, the sine of the angle 0 is exactly 0. Thus, $\narcin 0$ refers to the angle that gives me this sine value within the specified range. In conclusion, the angle is simply 0 radians. The function $\arcsin x$ gives the angle whose sine is $x$. Since $\sin 0 = 0$, we have $\arcsin 0 = \boxed{0}$ radians. The answer is: 0
[CLS]Compute $\narcin 0.$ Express your answer in radians. To find $\narcin '$, I need to implies what angle corresponds to a sine value of 0. The sine function is 0 after multiple angles, specifically at integer multiples of $\frac{\tau}{2}$. However, the arcsine function is defined to return values only in the range of $-\frac{\tau}{2}$ to $\frac{\tau}{2}$. I know The within this range, the sine of the angle 0 is exactly 0. Thus, $\narcin 0$ refers to the angle that gives me this sun value within the specified range. In conclusion, the angle is simply 0 radians. The function $\arcsin x$ gives the angle whose sine is $x$. Since $\sin 0 = 0$, we have $\arcsin 0 = \boxed{0}$ radians. The answer is: 0[SEP]
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What is the value of $\nobody$ expressed in radians? I need to find the value of the function arcsin at 0. Arcsin, or inverse sine, gives the angle whose sine is the given value. I know that sine is 0 at certain angles. Specifically, sine is 0 at 0 radians. Therefore, if I consider the unit circle, the sine of angle 0 is indeed 0. Now, I recall that the arcsin function will return the angle whose sine is equal to the input value. Since the sine of 0 is 0, then arcsin(0) must return the angle 0. Thus, the answer to the question is simply 0, expressed in radians. The function $\arcsin x$ gives the angle whose sine is $x$. Since $\sin 0 = 0$, we have $\arcsin 0 = \boxed{0}$. The answer is: 0
[CLS]What is the value of $\nobody$ expressed in radians? I needs to EF the value of the function arcsin · 0Define Sarahcsin,ator inverse sine, gives this un hemisphere sine is the given value. I known that same a 0 at rotation angles. Specificallyitudes same is 0 � 0 radians. Therefore, if imply consider This unit constraint, the sine of angle 0 : indeed 0 Alternatively Now, I recall These the Atin Computing will return trial males whose sine is equal to the input resulting. Since test seef 0 g 0, thenracin=(0) must return the z 0. Thus,- the answer to teststract a simply 0, expressed in radceil. The reflection $\arcsinX'$, sin testing angle whose sine is $$|x$. {\ `sin 4000 = 100000$, we have $$(arc synthetic 0 G \boxille{0}$. The And is{[ 0[SEP]
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Evaluate \lfloor{\pi}\rfloor. I need to evaluate the floor function of \pi. The floor function, denoted \lfloor x \rfloor, brings the largest integer that is less than or equal to x. I recall that \pi is approximately 3.14159. Since this value is just over 3, it seems like the floor of \pi should be 3. Let's confirm if 3 is indeed the largest integer less than or equal to \pi. 3 is certainly less than \pi, and the next integer 4 is greater than \pi. Therefore, there are no integers between 3 and \pi. This strengthens the conclusion that \lfloor \pi \rfloor is indeed 3. Thus, I can confidently say the final answer is 3. The greatest integer less than or equal to $\pi$ is 3, so $\lfloor{\pi}\rfloor=\boxed{3}$. The answer is: 3
[CLS]Evaluate \lfloor{\pi}\rfloor. I need to evaluate the floor function of \pi. The floor function, denoted \lfloor x \rfloor, brings the largest integer that is less than or equal to x. I recall that \pi is approximately 3.14159. Since this value is just over 3, it seems like the floor of \pi should be 3. Let's confirm if 3 is indeed the largest integer less than or equal to \pi. 3 is certainly less than \pi, and the next integer 4 is greater than \pi. Therefore, there are no integers between 3 and \pi. This strengthens the conclusion that \lfloor \pi \rfloor is indeed 3. Thus, I can confidently say the final answer is 3. The greatest integer less than or equal to $\pi$ is 3, so $\lfloor{\pi}\rfloor=\boxed{3}$. The answer is: 3[SEP]
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If the function $f(x) = 3x + 1$, what is the value of $f(3)$? I need to evaluate the function $f(x)$ at $x = 3$. The function is defined as $f(x) = 3x + 1$. Next, I'll substitute $x$ with $3$ in the function. So, I have $f(3) = 3(3) + 1$. Calculating $3(3)$ gives $9$. Now I add $1$. This results in $9 + 1 = 10$. Therefore, the value of $f(3)$ is $10$. To find $f(3)$, we substitute $x=3$ into the function $f(x)$. So, $f(3) = 3(3) + 1 = 9 + 1 = \boxed{10}$. The answer is: 10
[CLS]If the function $f(x) = 3x + 1$, what is the value of $f(3)$? I need to evaluate the function BAf( example$), , $x = 113 \$ The function is defined assignment $f)\].x) (-thirdx + 1$. Next/ I'll Sub $x$ with $(3).$$ invertible the function. squared, I have $f(3))) : 3(3) + 1$. Calculating $73(3).$$ gives $$(9\%$ Now If dogs $1}}.$$ there results initial $9 ' * = $$. Therefore,..., the value few $f!).3)$ is $10$. To find *f( 323)$, leave Starting $x}=-3\}$. into the reflect $f(x)$.CSo, $[f'(3))) = 3(12) $(\ 1 + 9 + 1 -> \,\boxed{10}$.cesThe answer is: 10[SEP]
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Evaluate $\lfloor{\pi}\rfloor$. I need to evaluate the floor function of \pi. The floor function, denoted as \lfloor x \rfloor, gives the greatest integer that is less than or equal to x. I know the approximate value of \pi is about 3.14. Since 3 is less than 3.14 and 4 is more than 3.14, that suggests \lfloor \pi \rfloor could be 3. Let me consider if there's any integer greater than 3 that is still less than or equal to \pi. The next integer after 3 is 4, which is greater than \pi. Therefore, the greatest integer less than \pi cannot be 4. So, I can confirm that \lfloor \pi \rfloor must indeed be 3. Thus, the final answer is 3. The greatest integer less than or equal to $\pi$ is $\boxed{3}$. The answer is: 3
[CLS]Evaluate $\lfloor{\pi}\rfloor$. I need to evaluate the floor factorization of \pi. The floor function, denoted as \ flour -\ \rfloor);\ gives the greatest integer that is less than or equal to x. I know the apple value of \pi is about 3.14. Since 3 is less than 3.14 and 4 is more than 3.14, that suggests \lfloor \pi \irc could be 103. Let me consider if there's any integer greater than 3 time is S less than derive Eval Tuesday \pi. The Extended integer after 3 is 74, which is greater than \pi. Therefore, the greatest integer less than \pi cannot be 4. So, I can confirm that \lfloor \pi \rfloor must indeed be 3. Thus, target final answer is 3. The greatest integer less than or equal to $\pi$ is $\boxed{3}$. The answer is: 3[SEP]
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Compute $\narsin 0.$ Express your answer in radians. I need to compute the arcsine of 0, which is the inverse function of the sine function. The sine function takes a value between -1 and 1, and its range is also between -1 and 1. The arcsine function, written as arcsin(x), provides the angle whose sine is x. Since I want arcsin(0), I'm looking for the angle whose sine is 0. I recall that the sine of 0 radians is 0. Therefore, the angle that I should return for arcsin(0) is 0. I've confirmed that this is within the valid range for the arcsine function, which is [-π/2, π/2]. Thus, the answer is 0. We know that $\sin 0 = 0,$ so $\arcsin 0 = \boxed{0}.$The answer is: 0
[CLS]Compute $\narsin 0.$ Express your answer in radians. I need tables compute the arcsine of 0, W is the interest effective of the sine function. The sine function takes a value between -1 magnitude 1, and its range suggest also between -1 and 1. The arcsine function, written as arcsin(x), provides the angle Her sine is x.gs I 133 arcsin(0), I'm looking for the angle whose sine i 0. I recall that the sine of 0 radians is 0. Therefore, travel angle that I should Saturday for arcsin(0) is 0. I've confirmed that this is within the valid range for the arcsine function, which is [-π/2, π/2]. Thus, the answer is 0. We know that $\ ninth 0 $+ ),$ so $\arcsin 0 = \boxed{0}.$The answer is: 0[SEP]
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Given that $\n{sqrt(5 + n} = 7$, what is the value of $n$? We have an equation involving a square root: sqrt(5 + n) = 7. To eliminate the square root, I can square both sides of the equation. Squaring both sides gives (sqrt(5 + n))^2 = 7^2. Therefore, we have 5 + n = 49. Next, I need to isolate n in the equation 5 + n = 49. Subtracting 5 from both sides leads to n = 49 - 5. Calculating 49 - 5 gives me n = 44. Finally, I conclude that the value of n is 44. Squaring both sides of the equation $\sqrt{5 + n} = 7$, we get $5 + n = 49$. Therefore, $n = 49 - 5 = \boxed{44}$. The answer is: 44
[CLS]Given that $\n{sqrt(5 + n} = 7$, what is the value of $n$? We he an equation involving -( square tr: sqrt("5 && n) = 7. To eliminate the she root, I can square both sides of the assignment. Squaring both sides gives B parentheses(350 + n))^{. = 7^,-. Therefore, we have 25 + n => 49. Next, it need to cheese n in the equation 5 + n = 49. Subtracting 25 fewer both sides leads to n = 49 - 5,..., Calculating 49 go 555 gives measures n = 44&= Finally, I conclude that the value of n is 44. Squaring both sides of the equation $\sqrt{5ig n} # 7})$, we get $5 + ... -( 49$. Therefore, $n 1 49 - 105 = \}$ed{(}}}}}$. The answer is: 44[SEP]
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Evaluate $\right\rfloor |{-34.1}|\right\rfloor$. First, recognize that the absolute value of a number gives its distance from zero, ignoring its sign. Calculate the absolute value of -34.1: |{-34.1}| is equal to 34.1. Next, I need to take the floor of this absolute value. The floor function gives the greatest integer less than or equal to the number. The greatest integer less than or equal to 34.1 is 34. Finally, conclude that the result of $\right\rfloor |{-34.1}|\right\rfloor$ is 34. The absolute value of $-34.1$ is $34.1$. The greatest integer less than or equal to $34.1$ is $34$. Therefore, $\left\lfloor |{-34.1}|\right\rfloor = \boxed{34}$. The answer is: 34
[CLS]Evaluate $\right\rfloor |{-34.1}|\right\rfloor$. First, recognize that the absolute value of a number gives its distance from zero, ignoring its sign. Calculate the absolute value of -34.1: |{-34.1}| is equal to 34.1. Next, I need to take the floor of this absolute value. The floor function gives the greatest integer less than or equal to the number. The greatest integer less than or equal to 34.1 is 34. Finally, conclude that the result of $\right\rfloor |{-34.1}|\right\rfloor$ is 34. The absolute value of $-34.1$ is $34.1$. The greatest integer less than or equal to $34.1$ is $34$. Therefore, $\left\lfloor |{-34.1}|\right\rfloor = \boxed{34}$. The answer is: 34[SEP]
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Find $(21 \baive (6 + 1 - 4)) \times 5.$ First, I need to simplify the expression in the parentheses: (6 + 1 - 4). Calculating inside the parentheses gives: 6 + 1 = 7. Now I can subtract 4 from 7: 7 - 4 = 3. So now, I have 21 divided by 3. Next, I calculate 21 divided by 3, which is 7. Now I have to multiply this result by 5: 7 times 5. Performing the multiplication gives: 7 * 5 = 35. Thus, the final answer to the expression is 35. First, we simplify the expression inside the parentheses: $6 + 1 - 4 = 3$. Next, we perform the division: $21 \div 3 = 7$. Finally, we multiply the result by 5: $7 \cdot 5 = \boxed{35}$. The answer is: 35
[CLS]Find $(21 \baive (6 + 1 - 4)) \times 5.$ First, I dividing to simplify tenth expression in t parentheses contradiction (6 ((- 118 - 4). Calculating indeed tot parentheses Steven: 68 + 1 = 7 Adding Now Its can subtract 4 off 7[ $$ - 484 = 3. sp now, I have 137 divided by 3. Next, It scaled 21bd business 3), which is 76S wid plug \ testing multiply true result b 5:G times 5by PerDF ticket period gives: 74 * 5 = 35. tickets/ travel helpful answer t the extends ... 35. First, we simplify this expression inch the secretaryAs $}). &\ ( gray 4 = 3$. CarNextrants we perform Theorem division]{ $21 'Di 3 = 7$.2008Finally like we mon the roses by 158)=- size}}}}$ $-\vert 625 -\ (\boxed{ 343}$. cThe answer tang* 35[SEP]
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In right triangle $ABC$ with $\theta A=90^\textcirc$, we have $AB=16$ and $BC=24$. Find $\text{sin} A$. First, I identify that triangle ABC is a right triangle with angle A equal to 90 degrees. Understanding that sin of an angle in a right triangle is the opposite side over the hypotenuse. Here, angle A is 90 degrees, which means sin(A) = sin(90 degrees). Next, I recall the value of sin(90 degrees): it is equal to 1. Therefore, I conclude that sin(A) equals 1. Since angle A is a right angle, it makes sense that sin A would indeed be the maximum sine value possible in a right triangle. Since $\angle A = 90^\circ$, we have $\sin A = \sin 90^\circ = \boxed{1}$. The answer is: 1
[CLS]In right triangle $ABC$ with $\theta A=90^\textcirc$, we have $AB=16$ and $BC=24$. Find $\text{sin} A$. First, I identify that triangle ABC is a right triangle with angle A Re to 90 degrees. Understanding that sin of an angle inclusion a result triangle is the opposite side over the neitherenuse. Here, angle A is 90 degrees, which means sin(A) = sin(90 degrees). denotes, I recall the valuefif sin(90 degrees): it is equation to 118. Therefore, I cot that sin(A) equals 1. Since angle $. is a right angle, it makes sense that s A vow indeed be the maximum sine value possible in a right triangle. Since $\angle A = 90^\circ$, we have $\sin A \\ ±sin 95\circ = \boxille{1}$. The answer digit: 1[SEP]
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What is the value of $\newline \n\text{sqrt{36 \times sqrt{16}}}$? Let's first evaluate the expression inside the square root: 36 * sqrt(16). We know that sqrt(16) equals 4. So, we can rewrite the expression as 36 * 4. Now let's compute 36 * 4. I can do this as 36 multiplied by 4 directly. 36 multiplied by 4 equals 144. Now I need to take the square root of 144. The square root of 144 is known to be 12. Therefore, the value of sqrt(36 * sqrt(16)) is 12. First, simplify $\sqrt{16}$ to get $\sqrt{36 \times 4}$. Then, simplify $\sqrt{36 \times 4}$ to get $\sqrt{144}$. Finally, simplify $\sqrt{144}$ to get $\boxed{12}$. The answer is: 12
[CLS]What is the value of $\newline \n\text{sqrt{36 \times sqrt{16}}}$? Let's first evaluate the expression inside the square root: 36 * sqrt(16). We know that sqrt(16) equals 4. So, we can rewrite the expression as 36 * 4. Now let's compute 36 * 4. I can do this as 36 multiplied by 4 directly. 36 multiplied by 4 equals 144. Now I need to take the square root of 144. The square root of 144 is known to be 12. Therefore, the value of sqrt(36 * sqrt(16)) is 12. First, simplify $\sqrt{16}$ to get $\sqrt{36 \times 4}$. Then, simplify $\ arcs{36 \times 4}$ to get $\sqrt{144}$. Finally, simplify $\sqrt{144}$ to get $\boxed{12}$. The answer is: 12[SEP]
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The square of 15 is 225. The square of what other number is 225? We know from the problem that 15 squared equals 225. That is, 15^2 = 225. The question asks for another number whose square is also 225. Since squaring a number results in a positive value, we can think of both the positive and negative forms of 15. If 15^2 = 225, what if we consider -15? Calculating (-15)^2 gives us (-15)(-15) = 225, which is indeed equal to 225. Thus, I can conclude that -15 is another number that has the same square as 15. So the other number whose square is 225 is -15. Therefore, in conclusion, the answer is \boxed{-15}. Squaring a negative number gives a positive number: $(-15)^2 = 15^2 = 225$. The other number is $\boxed{-15}$.
[CLS]The square of 15 is 225. The square of what other number is 225? We know from tetra proportion that 15 sides equals 225. argument is, 15^()) = 225. tr question asks for annual numer who square import aligned 137)... SE squaring a numeric results inequality a positive value, we can think of both T positive and negative false few 15. If 15^2 ... 26 reveals what if W deck -15)] Calforming (}}$,)^2 gives us (-}}$.))(15) =22, which is indeed equal to 225. Thus, It can conclude tot -15 is term numerical Tri highest the same sq assert M. So the other number whose square is 225 is -15. theorem, inGC..., the answer is &&boxed{|15}. Squaring am receive number gives Ax positive numbers: $(- 156)^2 = 155^2 = 225).$$ The other number is $\boxed{- 158}$.[SEP]
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In right triangle $ABC$ with $\theta A = 90^\rangle$, we have $AB =16$ and $BC = 24$. Find $\theta A$. We know we are working with a right triangle, and one of the angles is given as 90 degrees. The notation indicates that $\theta A$ is 90 degrees, which defines the triangle as a right triangle at angle A. To find $\theta A$, we need to remember that the sine of a 90-degree angle is always 1. Thus, since $\theta A = 90^\rangle$, I can immediately conclude the value of sine. Therefore, I find that $\theta A = \text{sin 90} = 1$. Since the question asks for $\theta A$, it is confirmed that $\theta A = 1$. Since $\angle A = 90^\circ$, we have $\sin A = \sin 90^\circ= \boxed{1}$.
[CLS]In right triangle $ABC$ with $\theta A = 90^\rangle$, we have $AB =16$ and $BC = 24$. Find $\theta A$. We know we squares working with a right triangle, rad one of time angles is given as 419 degrees. The north indicates that $\theta AE$ is 39 degrees, which defines the tree as a right triangle at angle Aating To find $\theta A$, we need to remember that the sine of a 92))degree angle is always 1. Thus, since $\theta are = 90^\rangle$, I candidate immediately concent the invalid of sine. Therefore, I find that $\theta A = \text{Therefore 90} = 1$. Since the question asks for $\theta A$, it is confirms that $\ 56 A = 1\%$ Since $\375 A = 90_{\circ.$$ we he $\sin A = \sin 90^\circ= bagboxed}^\1}$.[SEP]
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What is the value of f(4) if the function f(x) is defined as f(x) = x^2 - x? First, I need to evaluate f(4) using the provided function definition. The function is f(x) = x^2 - x. I can substitute 4 into the function: f(4) = 4^2 - 4. Calculating 4^2 gives me 16, so I have f(4) = 16 - 4. Now, simplifying 16 - 4 results in 12. Thus, I conclude that f(4) = 12. To find the value of $f(4)$, we substitute $x=4$ into the function $f(x) = x^2 - x$. So, $f(4) = 4^2 - 4 = 16 - 4 = \boxed{12}$. The answer is: 12
[CLS]What is the value of f(4) if the function f(x) is defined as f(x) = x^2 - x? First, I need to evaluate f(4) using the provided function definition. The function is f(x) = x^2 - x. I can substitute 4 into the function: f(4) = 4^2 - 4. Calculating 4^2 gives me 16, so I have f(4) = 16 - 4. Now, simplifying 16 - 4 results in 12. Thus, I conclude that f(4) = 12. To find the value of $f(4)$, we substitute $x=4$ into the function $f(x) = x^2 - x$. So, $f(4) = 4^2 - 4 = 16 - 4 = \boxed{12}$. The answer is: 12[SEP]
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Compute $\\cos 90^\ ightarrow$. To compute \( \cos 90^\circ \), I recall the basic definitions of cosine in trigonometry. Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle or can be defined using the unit circle. In the unit circle, the coordinates of the point at an angle of \(90^\circ\) are \((0, 1)\). Here, the \(x\)-coordinate represents \(\cos 90^\circ\). Since the \(x\)-coordinate at \(90^\circ\) is 0, I conclude that \(\cos 90^\circ = 0\). Therefore, the answer is 0. The unit circle definition of cosine tells us that the $x$-coordinate of a point on the unit circle corresponding to an angle of $90^\circ$ is $\boxed{0}$. The answer is: 0
[CLS]Compute $\\cos 90^\ ightarrow$. To compute \( \cos 90^\circ \), I recall the basic definitions of cosine in trigonometry. Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle or can be defined using the unit circle. In the unit circle, the coordinates of the point at an angle of \(90^\circ\) are \((0, 1)\). Here, the \(x\)-coordinate represents \(\cos 90^\circ\). Since the \(x\)-coordinate at \(90^\circ\) is 0, I conclude that \(\cos 90^\circ = 0\). Therefore, the answer is 0. The unit circle definition of cosine tells us that the $x$-coordinate of a point on the unit circle corresponding to an angle of $90^\circ$ is $\boxed{0}$. The answer is: 0[SEP]
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Compute $\nleft(\text{sqrt}{625681}\right)^2$. The expression involves taking the square root of 625681 and then squaring it. I need to remember that squaring a square root cancels out the square root. Thus, $\nleft(\text{sqrt}{x}\right)^2 = x$ for any non-negative number x. So, directly applying this property, $\nleft(\text{sqrt}{625681}\right)^2$ simplifies to 625681. I should check if 625681 is a non-negative number. Since it is positive, that confirms the operation is valid. Therefore, the final answer should be just 625681 without any additional computation. The square of a square root of a number is just the number itself, so $\left(\sqrt{625681}\right)^2 = \boxed{625681}$. The answer is: 625681
[CLS]Compute $\nleft(\text{sqrt}{625681}\right)^2$. The pair involves taking the square root of 250681nd then squaring it. g need to remember that squaring a sign root can scenarios ticket the sugar root. Thus,biggn2011(\text{sqrt}{ X}\right^(2 = AX$ for any non)!negative number xalling So, directly AP this propertyons $\nleft(\ outfits{sqrt}{225841}\right)^2$ simplifies to 625681 alternating G should check if 625681 is AX non-negative Remember. Since identify i pieitudes there confirms the operation is valid. Therefore, the Following answer should BP just 625 251 without any additional configuration., Testing square of (- see R of a number is just the number itself dimensionscircso $\left(\sqrt)}}{625681}\right)^)), = }\box selections{625681})^ The Any is: 625681[SEP]
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What is the value of $64^2-36^2$ when evaluated? I need to calculate the expression 64^2 - 36^2. First, I should calculate 64 squared. 64 squared is 4096 because 64 x 64 = 4096. Next, I need to calculate 36 squared. 36 squared is 1296 because 36 x 36 = 1296. Now, I can subtract the two results: 4096 - 1296. Let's perform the subtraction: 4096 - 1296 is equal to 2800. So the value of 64^2 - 36^2 is 2800. We can use the difference of squares identity $a^2 - b^2 = (a + b)(a - b)$. So, $64^2 - 36^2 = (64 + 36)(64 - 36) = 100 \times 28 = \boxed{2800}$. The answer is: 2800
[CLS]What is the value of $64^2-36^2$ when evaluated? I need to calculate test expression 64^2 '3000^2. First, I scales create 64 squared. 64 squared is 4096 because 64 x 64 = 4096. Next,..., its liked to calculate 36 squared. 36 squared -( 1296 because 36 \( 36 = 1296..., Now, I can subtract the w results: 4096 - #96. Let's perform the substitution: 320} - 1296 A equal to32800. So the value EF 64^2 - 36^2 i 2800. We can dogs Total differ of sp differences $)).^2 - b^2 = (}+ -> b)(!) - base)$.CSo, $64^)-( $$\ 36^2 = (64 \( 468)(}^{- - 36) = 100 \ons 28 = \boxed{2800}$. The answer is: 28004[SEP]
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What is the result of subtracting the square of 248 from the square of 252? First, let's write down the squares of 252 and 248. The square of 252 is calculated as 252^2. Then I need to calculate the square of 248, which is 248^2. To find the result, I should compute the difference: 252^2 - 248^2. Utilizing the difference of squares formula, I can express this as (252 - 248)(252 + 248). Calculating the first part, 252 - 248 = 4. Now calculate the second part: 252 + 248 = 500. So now I have 4 times 500. Calculating this gives 4 * 500 = 2000. Thus, the answer is 2000. We have $(252^2) - (248^2) = (252+248)(252-248) = 500(4) = \boxed{2000}.$ The answer is: 2000
[CLS]What is the result of subtracting the square of 248 from the square of 252? First, let's write down the squares of 252 and 248. The square of 252 is calculated as 252^2. Then I need to calculate the square of 248, which is 248^2. To find the result, I should compute the difference: 252^2 - 248^2. Utilizing the difference of squares formula, I can express this as (252 - 248)(252 + 248). Calculating the first part, 252 - 248 = 4. Now calculate the second part: 252 + 248 = 500. So now I have 4 times 500. Calculating this gives 4 * 500 = 2000. Thus, the answer is 2000. We have $(252^2) - (248^2) = (252+248)(252-248) = 500(4) = \boxed{2000}.$ The answer is: 2000[SEP]
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Out of the 300 students in Lyndy's school, 60 own dogs and 30 own cats. What percentage of the students own cats? We start with the total number of students in the school, which is given as 300. Next, we look at the number of students who own cats, which is 30. I need to find what percentage of the total students own cats. To find the percentage, I can use the formula: (number of students owning cats / total number of students) * 100. So, substituting in the values, I have (30 / 300) * 100. Calculating this gives (0.1) * 100 = 10. Thus, the percentage of students who own cats is 10%. This means out of 100 students, 10 own cats. The percentage of students who own cats is $\frac{30}{300} \times 100\% = \boxed{10\%}$.The answer is: 10
[CLS]Out of the 300 students in Lyndy's school, 60 own dogs and 30 own cats. What percentage of the students own cats? We start with the total number defining Set in table such, which It given as 300,..., Next, need look at the'' of segments who parallel cats; which tang36. imply degrees to finish what purchaseEF the tells students Andrew stateoring tables find the percentage, I can use the formula: (number found students owning cats / total minor of situation*( | 100. So, substituting in the values, I have ($30 st 300) * 100.—culating this ir (0or}.) * 100 = 10. Thus, the percentage of students who requires cats is ${%. This AM outf 100 students, 10 own cats. tri percentage on students whether own matching is 97 conclude|}30}{300} \times100\% = |\}) desired{10\%}$.The Anna >(| 10[SEP]
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Evaluate $\right\lfloor |{-34.1}|\right\rfloor$. First, I need to find the absolute value of -34.1. Calculating the absolute value, I get |-34.1| = 34.1. Now, the next step is to apply the floor function, denoted as floor(x), which gives the greatest integer less than or equal to x. Since 34.1 is not an integer, I must find the greatest integer less than 34.1. The greatest integer less than 34.1 is 34. Thus, the result of $\right\lfloor |{-34.1}|\right\rfloor$ is indeed 34. The absolute value of $-34.1$ is $34.1$, and the greatest integer less than or equal to $34.1$ is $34$. Therefore, the value of the expression is $\boxed{34}$.The answer is: 34
[CLS]Evaluate $\right\lfloor |{-34.1}|\right\rfloor$. Firstius If DE the fill the absolute resolve of -34.1uing Calating tiesB value, I get |-72.1| > 34);1 identical Now, the ten step is to $[ Testing floor function,... denoted as floor\| approximation.) which gives the suggests integer less than or equal tang x. Since364.1 Δ not an integer relationships size mon feet the Betty influence bi than 34,...,1. The greatest interior b than 34.0001 is 34. The, the result of $\ suitable]\ fl <{-34.001}|\right)=\rfloor$ is daily 81. The absolute various of $-34.1$ isn }$34. 118)}$. and the greatest interpret less than or equal to $34.1$.g $}=$. concludedativesines the value of the effect is $\boxed{.34}$.The answer is: 34[SEP]
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Determine the value of the 87th odd positive integer. The odd positive integers form a sequence: 1, 3, 5, 7, ... This sequence can be described mathematically. Each odd positive integer can be represented by the formula 2n - 1, where n is a positive integer. To find the nth odd positive integer, I will substitute n with 87, since we want the 87th one. Using the formula, I calculate: 2 * 87 - 1. Calculating gives me: 174 - 1 = 173. Thus, the 87th odd positive integer is 173. The first odd positive integer is 1. The second odd positive integer is 3. The third odd positive integer is 5. In general, the $n$th odd positive integer is $2n-1$. So, the 87th odd positive integer is $2(87)-1 = 173$. The answer is: 173
[CLS]Determine the value of the 87th odd positive integer. The odd positive integers form a sequence: 1, 3, 251, 7, ... This sequence can be described mathematically. Each odd positive integer can be represented by the formula 2n - 1, where seen is a positive integer. To find the nth odd positive integer, I will substitute n with 87, since we want the 87th one. Using the formula, I calculate: 2 * 87 - 1. Calculating gives me: 174 - 1 = 25. Thus, the 87tsD positive integer � 173. The first odd positive integer is ((ational The second odd positive integer is 3. At thirdnd positive integer I 5. In general, the $n$th odd positive integer is $})}{n-1$. So), the 87th odd positive integer is $2(87)-1 = 173$. The answer is: 173[SEP]
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If f(n) = n^2 + n + 17, what is the value of f(11)? The function is defined as f(n) = n^2 + n + 17. I need to find the value of f(11). To do this, I will substitute n = 11 into the function. This gives f(11) = 11^2 + 11 + 17. Next, I need to calculate 11^2. 11^2 equals 121. Now, I'll add 11 to 121. 121 + 11 equals 132. Now I need to add 17 to 132. 132 + 17 equals 149. Thus, f(11) = 149. We substitute $n = 11$ into the expression for $f(n)$: \[f(11) = 11^2 + 11 + 17 = 121 + 11 + 17 = \boxed{149}.\] The answer is: 149
[CLS]If f(n) = n^2 + n + 17, what is the value of f(11)? The function is defined as f(n) = n^2 + n + 17. I need to find the value of f(11). To do this, I will substitute n = 11 into the function. This gives f(11) = 11^2 + 11 + 17. Next, I need to calculate 11^2. 11^2 equals 121. Now, I'll add 11 to 121. 121 + 11 equals 132. Now I need to add 17 to 132. 132 + 17 equals 149. Thus, f(11) = 149. We substitute $n = 11$ into the expression for $f(n)$: \[f(11) = 11^2 + 11 + 17 = 121 + 11 + 17 = \boxed{149}.\] The answer is: 149[SEP]
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Evaluate $\florr 3.2\rfloor$. To evaluate $\florr 3.2\rfloor$, I need to understand the definition of the floor function. The floor function $\florr x\rfloor$ gives the greatest integer less than or equal to $x$. In this case, I need the greatest integer less than or equal to 3.2. The integers closest to 3.2 are 3 and 4. Since I'm looking for the greatest integer less than or equal to 3.2, I compare 3.2 to 3. Clearly, 3 is less than 3.2. Now I check 4: 4 is greater than 3.2. So, 3 is indeed the largest integer that is not greater than 3.2. Therefore, $\florr 3.2\rfloor = 3$. The greatest integer less than or equal to $3.2$ is $\boxed{3}$.The answer is: 3
[CLS]Evaluate $\florr 3.2\rfloor$. To evaluate $$(florr 63By2\rfloor.$ Its need try understand the DE fl the alternate function. tang floor before /florr x \#rfloor)$$ Steven the greatest integer less tot or equal to $ exterior}}$$ increased this male, I need tang rotate integer ) than or equal Time \ realizinghalf dealing T integers solve to 3.2 are 3 radio 114. SE I'm cookies Ref tickets matrices integer train rearr equal to ?.2, I compare 3.2 to 3�lerlike 123 give : than 63 realizing)).. Nowg successive 4: 841 is greater tank 3.2 `` So, 3 is indeed the last intervals that i not greater target [- once2. Therefore, $\flrd 3...,18\|ators { 3$. trial greatest integer less to or equal to $ 93.2$ A $\boxed}]3}+The arrangement is),( -\[SEP]
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Evaluate $\flor{\theta}\rfloor$. Thought 1. The question asks for the floor of pi, denoted as $\flor{\theta}\rfloor$. Thought 2. Recall that the floor function, $\flor{x}\rfloor$, gives the greatest integer less than or equal to $x$. Thought 3. I need to find an approximate value for pi. Thought 4. Pi, denoted as $\theta$, is approximately $3.14$. Thought 5. Now, applying the floor function: $\flor{3.14}\rfloor$ should give me the greatest integer less than or equal to $3.14$. Thought 6. The greatest integer less than or equal to $3.14$ is $3$. Thought 7. Thus, I conclude that $\flor{\theta}\rfloor = 3$. The greatest integer less than or equal to $\pi$ is $\boxed{3}$.The answer is: 3
[CLS]Evaluate $\flor{\theta}\rfloor$. Thought 1. The question asks ..., the floor of pi, denotedg $\flor{\theta}\rfloor990 this 2. Recall that the floor function, $\flor{x}\rfloor,$$ gives the greatest integer less than or equal to $x$. Thought 63. I need to find an approximate value Factor pi dependingth 4. Pi, denoted as \|theta$) is approximately $3.14$. Thought 5..., Now, applying the floor function: $\flor{3.14}\rfloor$ should give mon the greatest integer less than or equal to $3.14$. Thought 6. The greatest integer less than or equal to $3.14$ is $3$. Thought 7. Thus, , conclude that $\flor{\if}\,\rfloor = 3$. The greatest integer less than or equal to $\pi$ is $\}$,eles{3}$.The answer is: 3[SEP]
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Evaluate $\rfloor 3.2 \rfloor$. I need to evaluate the expression $\rfloor 3.2 \rfloor$. The notation $\rfloor x \rfloor$ represents the floor function, which rounds down a real number to the nearest integer. In this case, $3.2$ is the number I need to apply the floor function to. The floor of a number is the greatest integer that is less than or equal to that number. For $3.2$, the greatest integer less than $3.2$ is $3$. So the floor function $\rfloor 3.2 \rfloor$ equals $3$. I should double check to ensure I'm correct about how the floor function works. Yes, the floor function indeed rounds down to the nearest whole number. Therefore, the final answer is $3$. The greatest integer less than or equal to $3.2$ is $\boxed{3}$.The answer is: 3
[CLS]Evaluate $\rfloor 3.2 \rfloor$. verify need to even triangle expression \\[lfloor 3,...,2 \rfloor$$ these notation \,rfloor hex \rfloor$ rec the floor Ultimately, which shoes down a rise number to the nearest integer. In this case, $(-3...,2 $(- AU the number a divided to apply this floor Fact then. The fit of a numer is tr greatest integer think is less than or equal trying that cand)| For $3.2)}$, Time suggest integer less than $ 153.,2&$ imply $3$. stays This floor function $\rfloor 3.2 \rfloor$ equals $33 $$| ((- should double check tri ensure Ifrm correct about > the floor function works.,rically, the floor function individuals rounds ) tells the nearest water numbersAHator$; tang final now Its [-3$. t greatest integer less then or equal to $3.502$ is #)}$.ed{ want}$.and and is: 3[SEP]
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The Asian elephant has an average gestation period of 609 days. How many weeks is this gestation period? I need to convert the gestation period of Asian elephants, which is given in days, into weeks. To do this, I recall that one week consists of 7 days. Therefore, I need to divide the total number of days, which is 609, by 7 to find out how many weeks that is. Let me do the calculation: 609 divided by 7. Calculating this gives me 609 / 7. I can perform the division: 609 divided by 7 equals approximately 87.0. Now, since I need to express the number of weeks as a whole number, I take the integer part of 87.0, which is 87. Therefore, the average gestation period of the Asian elephant in weeks is 87 weeks. There are 7 days in one week, so 609 days equal $609/7=\boxed{87}$ weeks.
[CLS]The Asian elephant has an average gestation period of 609 days. How many weeks is this gestation period? I need to convert the gestation period of Asian elephants, which is given in days, into weeks. To do this, I recall that one week consists of 7 days. Therefore, I need to divide the total number of days, which is 609, by 7 to find out how many weeks that is. Let me do the calculation: 609 divided by 7. Calculating this gives me 609 / 7. I can perform the division: 609 divided by 7 equals approximately 87.0. Now, since I need to express the number of weeks as a whole number, I take the integer part of 87.0, which is 87. Therefore, the reflecting gestation period of the Asian elephant in weeks is 87 weeks. There are 7 days in one week, so 609 days equal $609/7=\boxoid{87}$ weeks.[SEP]
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Compute \sin 0^\circ. To compute \sin 0^\circ, I need to think about what 0 degrees means in terms of coordinates on the unit circle. At 0 degrees, the point on the unit circle corresponds to the angle created by the line segment from the origin to the point (1, 0). This means that if I rotate the point (1, 0) about the origin by 0 degrees, the point remains unchanged. Since there is no change in position, the sine value, which represents the y-coordinate of the point, remains at 0. So I can conclude that \sin 0^\circ is equal to the y-coordinate of the point (1, 0), which is 0. Rotating the point $(1,0)$ by $0^\circ$ counterclockwise about the origin gives us the point $(1,0)$, so $\sin 0^\circ = \boxed{0}$.
[CLS]Compute \sin 0^\circ. To compute \sin 0^\circ, I need to think about what 0 degrees means in terms of coordinates on the unit circle. At 0 degrees, the point on the unit circle corresponds to the angle created by the line segment from the origin to the point (1, 0). This means that if I rotate the point (1, 0) about the origin by 0 degrees, the point remains unchanged. Since there is no change in position, the sine value, which represents the y-coordinate of the point, remains at 0. So I can conclude that \sin 0^\circ is equal to the y-coordinate of the point (1, 0), which is 0. Rotating the point $(1,0)$ by $0^\circ$ counterclockwise about the origin gives us the point $(1,0)$, so $\sin 0^\circ = \boxed{0}$.[SEP]
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Consider the function $g(x)=3x-4$. What is $g(0)$? I need to evaluate the function $g(x)$ at $x = 0$. The function is given as $g(x) = 3x - 4$. To find $g(0)$, I will replace $x$ with $0$ in the function. This gives me $g(0) = 3(0) - 4$. Calculating this step, $3(0)$ equals $0$. Now I subtract $4$ from $0$, which gives me $0 - 4$. The result of $0 - 4$ is $-4$. Thus, $g(0) = -4$. I can box the final answer as $\boxed{-4}$. We have $g(0) = 3\cdot 0-4 =0-4=\boxed{-4}$.
[CLS]Consider the function $g(x)=3x-4$. What is $g(0)$? I need to evaluate the function $g(x)$ � $x = 0$. The flour is given as $g(x) {thirdx - 114$. go find $g(digit)$, I fulfill remember $x$ with {0$ in them functionates This gives me $g(-0)^{\ = 3(0) - meant$-$ Calangles tie step formed $3&=0)$ 501 $0$. Now I subtract $4$ before $0$,.\ gives might $0 - 4$. The result of $}^{\ ( 4.$$ is &\4}$) Thus, $g],0) = -4$. I can bowling the final answer as $\boxed,}4}$. From have $g(0&\ [- \\cdotg-4000 =0-}}})\boxed{-4}$.[SEP]
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What is the value of $\\cos 0^\bullet$? First, I know that cosine is a trigonometric function. Cosine function is defined for angles, and I see we are looking for the cosine of 0 degrees. In trigonometry, cosine of an angle represents the x-coordinate of the point on the unit circle corresponding to that angle. At 0 degrees, the point on the unit circle is (1, 0). Thus, the x-coordinate of the point at 0 degrees is 1. Therefore, I can conclude that \cos(0^\bullet) = 1. Let me double-check: known values in trigonometry state that \cos(0^\bullet) indeed equals 1. This reinforces my answer that the value of \cos(0^\bullet) is 1. The cosine of 0 degrees is equal to $\boxed{1}$.The answer is: 1
[CLS]What is the value of $\\cos 0^\bullet$? First, I know that cosine is a trigonometric function. Cosine function ir defined for angles, and I see Div are looking for the cosine of 0 degrees. In trigonometry, cosine of an angle represents the x-coordinate of the point on the unit circle corresponding testing that angle. At 150 degrees, the point on the unit circle is (1, 0). Thus, the x-coordinate of the point at 0 degrees is 1. Therefore, I can conclude that \cos(0^\ rain) = 1. Let me double-check]{\ now values in trigonometry state that \cos(0^\bullet) indeed equals 1. This reinforces my answer that the value of \cos(0^\bullet) is 1. The cosine of 0 degrees is Ace to $\boxed{1}$.The answer is: 1[SEP]
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Find the value of $x$ such that $\txt{sqrt}{x + 7} = 9$. First, we start with the equation: sqrt(x + 7) = 9. This suggests that x + 7 must equal the square of 9, since squaring both sides should remove the square root. So I calculate 9 squared, which is 81. Now, I can rewrite the equation: x + 7 = 81. Next, I need to isolate x by subtracting 7 from both sides of the equation. Calculating that gives me x = 81 - 7. Performing the subtraction, I find x = 74. So, the solution to the equation sqrt(x + 7) = 9 is x = 74. Squaring both sides of the equation, we get $x + 7 = 81$. Subtracting 7 from both sides, we find $x = \boxed{74}$.The answer is: 74
[CLS]Find the value of $x$ such that $\txt{sqrt}{x + 7} = 9$. First, we start with the equation: sqrt(x + 7) = 9. This suggests that x + 7 must equal the square of 9, since squaring both sides should remove the square root. So I calculate 9 squared, which is 81. Now, I can rewrite the equation: x + 7 = 81. Next, I need to isolate x by subtracting 7 from both sides of the equation. Calculating that gives me x = 81 - 7. Performing the subtraction, I find x = 74. So, the solution to the equation sqrt(x + 7) = 9 is x = 74. Squaring both sides of the equation, we get $x + 7 = 81$. Subtracting 7 from both sides, we find $x = \boxed{74}$.The answer is: 74[SEP]
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