prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Compute the quotient and remainder of 14329 divided by 72. Show the long division steps. | To divide 14329 by 72:
1. Determine how many times 72 goes into the leading digits of 14329.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 14329 / 72 = 199 with a remainder of 1.
Proof: (199 * 72) +... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 10^2 = c^2
2. Calculate the squares: 49 + 100 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 8^2 = c^2
2. Calculate the squares: 81 + 64 = c^2
3. Sum the squares: 145 = c^2
4. Take the square root of both sides: c = √145
Final Answe... |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[5, -4]
B =
[-5, 4]
[-1, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-5) + (1*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (-1*4) + (1*4) = 0
C21 = (A21 * B11) + (A22 * B21) = (5*-5) + (-4*-1) = -21
C22 = (A21 * B12) + (A22 * B22) = (5*4) + (-4*4) = 4
Result ... |
Compute the indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 5. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 5^2 = c^2
2. Calculate the squares: 289 + 25 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 15, 6, 14, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 15 + 6 + 14 + 15) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.8, 4.2, -4.8, 3.2, 4.2]
Squared Differences: [46.24, 17.64, 23.04, 10.24, 17.64]... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 10, 12, 17, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 10 + 12 + 17 + 2) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, 0.0, 2.0, 7.0, -8.0]
Squared Differences: [1.0, 0.0, 4.0, 49.0, 64.0]
Sum ... |
Add the fractions 11/14 and 3/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 9, which is 126.
2. Convert 11/14 by multiplying top and bottom by 9: 99/126.
3. Convert 3/9 by multiplying top and bottom by 14: 42/126.
4. Add the numerators: 99 + 42 = 141.
5. The resulting fraction is 141/126.
6. Simpl... |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[-4, 5]
B =
[5, 4]
[-1, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*5) + (-3*-1) = -12
C12 = (A11 * B12) + (A12 * B22) = (-3*4) + (-3*5) = -27
C21 = (A21 * B11) + (A22 * B21) = (-4*5) + (5*-1) = -25
C22 = (A21 * B12) + (A22 * B22) = (-4*4) + (5*5) = 9
Res... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Compute the quotient and remainder of 87624 divided by 21. Show the long division steps. | To divide 87624 by 21:
1. Determine how many times 21 goes into the leading digits of 87624.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 87624 / 21 = 4172 with a remainder of 12.
Proof: (4172 * 21... |
Compute the indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + C |
Compute the indefinite integral of f(x) = 3x^4. | To find the indefinite integral ∫ 3x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 15, 5, 19, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 15 + 5 + 19 + 5) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, 5.0, -5.0, 9.0, -5.0]
Squared Differences: [16.0, 25.0, 25.0, 81.0, 25.0]
S... |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Multiply the following 2x2 matrices:
A =
[-5, -3]
[0, -3]
B =
[-1, -3]
[4, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-1) + (-3*4) = -7
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (-3*-3) = 24
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (-3*4) = -12
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (-3*-3) = 9
R... |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Multiply the following 2x2 matrices:
A =
[4, -2]
[-1, 5]
B =
[-1, 5]
[5, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-1) + (-2*5) = -14
C12 = (A11 * B12) + (A12 * B22) = (4*5) + (-2*-4) = 28
C21 = (A21 * B11) + (A22 * B21) = (-1*-1) + (5*5) = 26
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (5*-4) = -25
Res... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 14. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 14^2 = c^2
2. Calculate the squares: 289 + 196 = c^2
3. Sum the squares: 485 = c^2
4. Take the square root of both sides: c = √485
Final A... |
Multiply the following 2x2 matrices:
A =
[-3, 2]
[-1, -5]
B =
[-5, 2]
[-2, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-5) + (2*-2) = 11
C12 = (A11 * B12) + (A12 * B22) = (-3*2) + (2*-2) = -10
C21 = (A21 * B11) + (A22 * B21) = (-1*-5) + (-5*-2) = 15
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (-5*-2) = 8
R... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 15^2 = c^2
2. Calculate the squares: 81 + 225 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[-3, -2]
B =
[0, -1]
[-5, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*0) + (0*-5) = 0
C12 = (A11 * B12) + (A12 * B22) = (-5*-1) + (0*-3) = 5
C21 = (A21 * B11) + (A22 * B21) = (-3*0) + (-2*-5) = 10
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (-2*-3) = 9
Resu... |
Multiply the following 2x2 matrices:
A =
[3, -4]
[5, -4]
B =
[-1, -3]
[-5, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*-1) + (-4*-5) = 17
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (-4*5) = -29
C21 = (A21 * B11) + (A22 * B21) = (5*-1) + (-4*-5) = 15
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (-4*5) = -35
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 4, 17, 2, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 4 + 17 + 2 + 15) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -6.2, 6.8, -8.2, 4.8]
Squared Differences: [7.84, 38.44, 46.24, 67.24, 23.04]
... |
Add the fractions 14/8 and 9/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 6, which is 24.
2. Convert 14/8 by multiplying top and bottom by 3: 42/24.
3. Convert 9/6 by multiplying top and bottom by 4: 36/24.
4. Add the numerators: 42 + 36 = 78.
5. The resulting fraction is 78/24.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[2, 5]
[4, 1]
B =
[4, -3]
[1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*4) + (5*1) = 13
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (5*3) = 9
C21 = (A21 * B11) + (A22 * B21) = (4*4) + (1*1) = 17
C22 = (A21 * B12) + (A22 * B22) = (4*-3) + (1*3) = -9
Result Matri... |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[-5, -2]
B =
[-4, -4]
[-5, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-4) + (-2*-5) = 6
C12 = (A11 * B12) + (A12 * B22) = (1*-4) + (-2*-5) = 6
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (-2*-5) = 30
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (-2*-5) = 30
R... |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[1, -3]
B =
[2, -3]
[-3, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*2) + (-5*-3) = 11
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (-5*4) = -14
C21 = (A21 * B11) + (A22 * B21) = (1*2) + (-3*-3) = 11
C22 = (A21 * B12) + (A22 * B22) = (1*-3) + (-3*4) = -15
R... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Add the fractions 2/13 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 2/13 by multiplying top and bottom by 1: 2/13.
3. Convert 5/13 by multiplying top and bottom by 1: 5/13.
4. Add the numerators: 2 + 5 = 7.
5. The resulting fraction is 7/13.
Final Answer: 7/13 |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 8, 14, 9, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 8 + 14 + 9 + 2) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, -0.6, 5.4, 0.4, -6.6]
Squared Differences: [1.96, 0.36, 29.16, 0.16, 43.56]
S... |
Add the fractions 11/4 and 12/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 6, which is 12.
2. Convert 11/4 by multiplying top and bottom by 3: 33/12.
3. Convert 12/6 by multiplying top and bottom by 2: 24/12.
4. Add the numerators: 33 + 24 = 57.
5. The resulting fraction is 57/12.
6. Simplify by d... |
Compute the quotient and remainder of 38423 divided by 97. Show the long division steps. | To divide 38423 by 97:
1. Determine how many times 97 goes into the leading digits of 38423.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 38423 / 97 = 396 with a remainder of 11.
Proof: (396 * 97) ... |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Add the fractions 2/2 and 13/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 15, which is 30.
2. Convert 2/2 by multiplying top and bottom by 15: 30/30.
3. Convert 13/15 by multiplying top and bottom by 2: 26/30.
4. Add the numerators: 30 + 26 = 56.
5. The resulting fraction is 56/30.
6. Simplify by... |
Add the fractions 6/13 and 8/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 12, which is 156.
2. Convert 6/13 by multiplying top and bottom by 12: 72/156.
3. Convert 8/12 by multiplying top and bottom by 13: 104/156.
4. Add the numerators: 72 + 104 = 176.
5. The resulting fraction is 176/156.
6. S... |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 11, 19, 16, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 11 + 19 + 16 + 15) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -2.8, 5.2, 2.2, 1.2]
Squared Differences: [33.64, 7.84, 27.04, 4.84, 1.44]
... |
Compute the quotient and remainder of 52485 divided by 85. Show the long division steps. | To divide 52485 by 85:
1. Determine how many times 85 goes into the leading digits of 52485.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 52485 / 85 = 617 with a remainder of 40.
Proof: (617 * 85) ... |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Compute the quotient and remainder of 52889 divided by 59. Show the long division steps. | To divide 52889 by 59:
1. Determine how many times 59 goes into the leading digits of 52889.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 52889 / 59 = 896 with a remainder of 25.
Proof: (896 * 59) ... |
Find the roots of the quadratic equation: 10x^2 + -10x + -2 = 0 | 1. Identify coefficients: a=10, b=-10, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -80 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √180) / 20.
5. Calculate roots: x1 = 1.170820393249937, x2 = -0.17082039324993695.
Fi... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 35215 divided by 30. Show the long division steps. | To divide 35215 by 30:
1. Determine how many times 30 goes into the leading digits of 35215.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 35215 / 30 = 1173 with a remainder of 25.
Proof: (1173 * 30... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[1, -4]
[-2, -2]
B =
[1, 5]
[4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*1) + (-4*4) = -15
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (-4*5) = -15
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (-2*4) = -10
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (-2*5) = -20
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 6, 9, 13, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 6 + 9 + 13 + 2) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.0, -4.0, -1.0, 3.0, -8.0]
Squared Differences: [100.0, 16.0, 1.0, 9.0, 64.0]
S... |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 19^2 + 3^2 = c^2
2. Calculate the squares: 361 + 9 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 1, 6, 10, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 1 + 6 + 10 + 13) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.4, -8.6, -3.6, 0.4, 3.4]
Squared Differences: [70.56, 73.96, 12.96, 0.16, 11.56]
... |
Compute the quotient and remainder of 6392 divided by 99. Show the long division steps. | To divide 6392 by 99:
1. Determine how many times 99 goes into the leading digits of 6392.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 6392 / 99 = 64 with a remainder of 56.
Proof: (64 * 99) + 56 ... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 9^2 = c^2
2. Calculate the squares: 256 + 81 = c^2
3. Sum the squares: 337 = c^2
4. Take the square root of both sides: c = √337
Final Ans... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 9^2 = c^2
2. Calculate the squares: 64 + 81 = c^2
3. Sum the squares: 145 = c^2
4. Take the square root of both sides: c = √145
Final Answe... |
Find the roots of the quadratic equation: 6x^2 + 3x + -6 = 0 | 1. Identify coefficients: a=6, b=3, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -144 = 153.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √153) / 12.
5. Calculate roots: x1 = 0.780776406404415, x2 = -1.2807764064044151.
Final A... |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Compute the quotient and remainder of 83346 divided by 30. Show the long division steps. | To divide 83346 by 30:
1. Determine how many times 30 goes into the leading digits of 83346.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 83346 / 30 = 2778 with a remainder of 6.
Proof: (2778 * 30)... |
Find the roots of the quadratic equation: 2x^2 + 2x + 1 = 0 | 1. Identify coefficients: a=2, b=2, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(2)(1) = 4 - 8 = -4.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √4i) / 4.
Final Answer: x = -0.5 ± 0.5i |
Add the fractions 11/7 and 9/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 5, which is 35.
2. Convert 11/7 by multiplying top and bottom by 5: 55/35.
3. Convert 9/5 by multiplying top and bottom by 7: 63/35.
4. Add the numerators: 55 + 63 = 118.
5. The resulting fraction is 118/35.
Final Answer: 1... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Find the roots of the quadratic equation: 6x^2 + -8x + 7 = 0 | 1. Identify coefficients: a=6, b=-8, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(6)(7) = 64 - 168 = -104.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √104i) / 12.
Final Answer: x = 0.6666666666666666 ± 0.8498365855987974i |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 10x^2. | To find the indefinite integral ∫ 10x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
Compute the quotient and remainder of 17084 divided by 77. Show the long division steps. | To divide 17084 by 77:
1. Determine how many times 77 goes into the leading digits of 17084.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 17084 / 77 = 221 with a remainder of 67.
Proof: (221 * 77) ... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 10, 4, 17, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 10 + 4 + 17 + 16) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, -3.0, -9.0, 4.0, 3.0]
Squared Differences: [25.0, 9.0, 81.0, 16.0, 9.0]
Su... |
Find the roots of the quadratic equation: 1x^2 + -5x + 8 = 0 | 1. Identify coefficients: a=1, b=-5, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(1)(8) = 25 - 32 = -7.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √7i) / 2.
Final Answer: x = 2.5 ± 1.3228756555322954i |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Add the fractions 8/8 and 13/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 8/8 by multiplying top and bottom by 15: 120/120.
3. Convert 13/15 by multiplying top and bottom by 8: 104/120.
4. Add the numerators: 120 + 104 = 224.
5. The resulting fraction is 224/120.
6. S... |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 18^2 = c^2
2. Calculate the squares: 324 + 324 = c^2
3. Sum the squares: 648 = c^2
4. Take the square root of both sides: c = √648
Final A... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[2, -2]
B =
[-5, 2]
[-4, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-5) + (-2*-4) = 33
C12 = (A11 * B12) + (A12 * B22) = (-5*2) + (-2*2) = -14
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-2*-4) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (-2*2) = 0
Res... |
Compute the quotient and remainder of 16821 divided by 44. Show the long division steps. | To divide 16821 by 44:
1. Determine how many times 44 goes into the leading digits of 16821.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 16821 / 44 = 382 with a remainder of 13.
Proof: (382 * 44) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 18, 11, 17, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 18 + 11 + 17 + 14) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.6, 5.4, -1.6, 4.4, 1.4]
Squared Differences: [92.16, 29.16, 2.56, 19.36, 1.96]
... |
Compute the quotient and remainder of 76367 divided by 77. Show the long division steps. | To divide 76367 by 77:
1. Determine how many times 77 goes into the leading digits of 76367.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 76367 / 77 = 991 with a remainder of 60.
Proof: (991 * 77) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 12, 7, 5, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 12 + 7 + 5 + 1) / 5 = 32 / 5 = 6.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, 5.6, 0.6, -1.4, -5.4]
Squared Differences: [0.36, 31.36, 0.36, 1.96, 29.16]
Su... |
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