prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
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 ... |
Add the fractions 3/10 and 15/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 2, which is 10.
2. Convert 3/10 by multiplying top and bottom by 1: 3/10.
3. Convert 15/2 by multiplying top and bottom by 5: 75/10.
4. Add the numerators: 3 + 75 = 78.
5. The resulting fraction is 78/10.
6. Simplify by di... |
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) |
Multiply the following 2x2 matrices:
A =
[3, 5]
[2, -5]
B =
[5, -2]
[-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) = (3*5) + (5*-5) = -10
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (5*4) = 14
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (-5*-5) = 35
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (-5*4) = -24
Resul... |
Multiply the following 2x2 matrices:
A =
[1, 2]
[-4, -2]
B =
[3, 1]
[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*3) + (2*1) = 5
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (2*-4) = -7
C21 = (A21 * B11) + (A22 * B21) = (-4*3) + (-2*1) = -14
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (-2*-4) = 4
Result M... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 10, 1, 7, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 10 + 1 + 7 + 7) / 5 = 29 / 5 = 5.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, 4.2, -4.8, 1.2, 1.2]
Squared Differences: [3.24, 17.64, 23.04, 1.44, 1.44]
Su... |
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) |
Find the roots of the quadratic equation: 7x^2 + 2x + 10 = 0 | 1. Identify coefficients: a=7, b=2, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(7)(10) = 4 - 280 = -276.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √276i) / 14.
Final Answer: x = -0.14285714285714285 ± 1.1866605518454392i |
Compute the quotient and remainder of 7950 divided by 44. Show the long division steps. | To divide 7950 by 44:
1. Determine how many times 44 goes into the leading digits of 7950.
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: 7950 / 44 = 180 with a remainder of 30.
Proof: (180 * 44) + 3... |
Add the fractions 6/10 and 1/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 2, which is 10.
2. Convert 6/10 by multiplying top and bottom by 1: 6/10.
3. Convert 1/2 by multiplying top and bottom by 5: 5/10.
4. Add the numerators: 6 + 5 = 11.
5. The resulting fraction is 11/10.
Final Answer: 11/10 |
Add the fractions 13/11 and 3/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 10, which is 110.
2. Convert 13/11 by multiplying top and bottom by 10: 130/110.
3. Convert 3/10 by multiplying top and bottom by 11: 33/110.
4. Add the numerators: 130 + 33 = 163.
5. The resulting fraction is 163/110.
Fin... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 12, 11, 1, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 12 + 11 + 1 + 15) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 4.0, 3.0, -7.0, 7.0]
Squared Differences: [49.0, 16.0, 9.0, 49.0, 49.0]
Sum... |
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 indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
Compute the quotient and remainder of 83721 divided by 10. Show the long division steps. | To divide 83721 by 10:
1. Determine how many times 10 goes into the leading digits of 83721.
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: 83721 / 10 = 8372 with a remainder of 1.
Proof: (8372 * 10)... |
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) = (5x + 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 = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 76959 divided by 96. Show the long division steps. | To divide 76959 by 96:
1. Determine how many times 96 goes into the leading digits of 76959.
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: 76959 / 96 = 801 with a remainder of 63.
Proof: (801 * 96) ... |
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 |
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 roots of the quadratic equation: 7x^2 + -3x + 1 = 0 | 1. Identify coefficients: a=7, b=-3, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(7)(1) = 9 - 28 = -19.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √19i) / 14.
Final Answer: x = 0.21428571428571427 ± 0.3113499245386196i |
Find the derivative of f(x) = (5x + 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 = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Compute the quotient and remainder of 88724 divided by 97. Show the long division steps. | To divide 88724 by 97:
1. Determine how many times 97 goes into the leading digits of 88724.
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: 88724 / 97 = 914 with a remainder of 66.
Proof: (914 * 97) ... |
Multiply the following 2x2 matrices:
A =
[5, 4]
[-1, -4]
B =
[3, 5]
[3, 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) = (5*3) + (4*3) = 27
C12 = (A11 * B12) + (A12 * B22) = (5*5) + (4*5) = 45
C21 = (A21 * B11) + (A22 * B21) = (-1*3) + (-4*3) = -15
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (-4*5) = -25
Result ... |
Multiply the following 2x2 matrices:
A =
[2, 1]
[2, -1]
B =
[-5, -2]
[-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) = (2*-5) + (1*-1) = -11
C12 = (A11 * B12) + (A12 * B22) = (2*-2) + (1*-5) = -9
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-1*-1) = -9
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (-1*-5) = 1
Res... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 17. 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 + 17^2 = c^2
2. Calculate the squares: 81 + 289 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Ans... |
Multiply the following 2x2 matrices:
A =
[0, 5]
[3, -5]
B =
[0, 1]
[-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) = (0*0) + (5*-4) = -20
C12 = (A11 * B12) + (A12 * B22) = (0*1) + (5*-3) = -15
C21 = (A21 * B11) + (A22 * B21) = (3*0) + (-5*-4) = 20
C22 = (A21 * B12) + (A22 * B22) = (3*1) + (-5*-3) = 18
Resul... |
Multiply the following 2x2 matrices:
A =
[0, 2]
[4, -5]
B =
[-5, -3]
[5, 0] | 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) = (0*-5) + (2*5) = 10
C12 = (A11 * B12) + (A12 * B22) = (0*-3) + (2*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (-5*5) = -45
C22 = (A21 * B12) + (A22 * B22) = (4*-3) + (-5*0) = -12
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 3, 16, 6, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 3 + 16 + 6 + 18) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -7.2, 5.8, -4.2, 7.8]
Squared Differences: [4.84, 51.84, 33.64, 17.64, 60.84]
... |
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 +... |
Add the fractions 2/6 and 11/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 13, which is 78.
2. Convert 2/6 by multiplying top and bottom by 13: 26/78.
3. Convert 11/13 by multiplying top and bottom by 6: 66/78.
4. Add the numerators: 26 + 66 = 92.
5. The resulting fraction is 92/78.
6. Simplify by... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(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) |
Compute the quotient and remainder of 99400 divided by 49. Show the long division steps. | To divide 99400 by 49:
1. Determine how many times 49 goes into the leading digits of 99400.
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: 99400 / 49 = 2028 with a remainder of 28.
Proof: (2028 * 49... |
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) |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^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: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 16, 8, 20, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 16 + 8 + 20 + 13) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, 2.4, -5.6, 6.4, -0.6]
Squared Differences: [6.76, 5.76, 31.36, 40.96, 0.36]
... |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^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: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Find the derivative of f(x) = (2x + 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 = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
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 |
Multiply the following 2x2 matrices:
A =
[-5, -1]
[-4, 3]
B =
[-5, 1]
[-5, 1] | 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) + (-1*-5) = 30
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (-1*1) = -6
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (3*-5) = 5
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (3*1) = -1
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 11, 11, 20, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 11 + 11 + 20 + 4) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, -0.2, -0.2, 8.8, -7.2]
Squared Differences: [1.44, 0.04, 0.04, 77.44, 51.84]... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 14, 1, 17, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 14 + 1 + 17 + 17) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.0, 4.0, -9.0, 7.0, 7.0]
Squared Differences: [81.0, 16.0, 81.0, 49.0, 49.0]
S... |
Find the roots of the quadratic equation: 8x^2 + -8x + -4 = 0 | 1. Identify coefficients: a=8, b=-8, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -128 = 192.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √192) / 16.
5. Calculate roots: x1 = 1.3660254037844386, x2 = -0.3660254037844386.
Final... |
Multiply the following 2x2 matrices:
A =
[4, -2]
[-3, -3]
B =
[4, -4]
[4, 0] | 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*4) + (-2*4) = 8
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (-2*0) = -16
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (-3*4) = -24
C22 = (A21 * B12) + (A22 * B22) = (-3*-4) + (-3*0) = 12
Resu... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 18, 13, 15, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 18 + 13 + 15 + 13) / 5 = 74 / 5 = 14.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, 3.2, -1.8, 0.2, -1.8]
Squared Differences: [0.04, 10.24, 3.24, 0.04, 3.24]
... |
Compute the quotient and remainder of 42281 divided by 52. Show the long division steps. | To divide 42281 by 52:
1. Determine how many times 52 goes into the leading digits of 42281.
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: 42281 / 52 = 813 with a remainder of 5.
Proof: (813 * 52) +... |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
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 indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^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: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 19. 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 + 19^2 = c^2
2. Calculate the squares: 81 + 361 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^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: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
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) |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^5 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: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Add the fractions 15/9 and 3/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 9, which is 9.
2. Convert 15/9 by multiplying top and bottom by 1: 15/9.
3. Convert 3/9 by multiplying top and bottom by 1: 3/9.
4. Add the numerators: 15 + 3 = 18.
5. The resulting fraction is 18/9.
6. Simplify by dividing... |
Compute the indefinite integral of f(x) = 2x^5. | To find the indefinite integral ∫ 2x^5 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: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 2 / 6 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^5 dx = 1/3x^6 + C |
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 |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 19. 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 + 19^2 = c^2
2. Calculate the squares: 289 + 361 = c^2
3. Sum the squares: 650 = c^2
4. Take the square root of both sides: c = √650
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 4. 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: 10^2 + 4^2 = c^2
2. Calculate the squares: 100 + 16 = c^2
3. Sum the squares: 116 = c^2
4. Take the square root of both sides: c = √116
Final Ans... |
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 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 |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 5^2 = c^2
2. Calculate the squares: 64 + 25 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[2, 4]
[0, 2]
B =
[0, -3]
[3, 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*0) + (4*3) = 12
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (4*3) = 6
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (2*3) = 6
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (2*3) = 6
Result Matrix ... |
Find the roots of the quadratic equation: 9x^2 + -4x + -3 = 0 | 1. Identify coefficients: a=9, b=-4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -108 = 124.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √124) / 18.
5. Calculate roots: x1 = 0.8408627069811135, x2 = -0.39641826253666906.
Fina... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 7. 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: 12^2 + 7^2 = c^2
2. Calculate the squares: 144 + 49 = c^2
3. Sum the squares: 193 = c^2
4. Take the square root of both sides: c = √193
Final Ans... |
Compute the quotient and remainder of 78039 divided by 79. Show the long division steps. | To divide 78039 by 79:
1. Determine how many times 79 goes into the leading digits of 78039.
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: 78039 / 79 = 987 with a remainder of 66.
Proof: (987 * 79) ... |
Add the fractions 5/7 and 1/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 9, which is 63.
2. Convert 5/7 by multiplying top and bottom by 9: 45/63.
3. Convert 1/9 by multiplying top and bottom by 7: 7/63.
4. Add the numerators: 45 + 7 = 52.
5. The resulting fraction is 52/63.
Final Answer: 52/63 |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) 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) = (5x + 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 = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 5^2 = c^2
2. Calculate the squares: 16 + 25 = c^2
3. Sum the squares: 41 = c^2
4. Take the square root of both sides: c = √41
Final Answer:... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[4, 2]
[-5, 2]
B =
[3, -1]
[-1, -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) = (4*3) + (2*-1) = 10
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (2*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (2*-1) = -17
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (2*-2) = 1
Resul... |
Find the roots of the quadratic equation: 10x^2 + 0x + -4 = 0 | 1. Identify coefficients: a=10, b=0, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -160 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √160) / 20.
5. Calculate roots: x1 = 0.6324555320336759, x2 = -0.6324555320336759.
Final ... |
Compute the quotient and remainder of 15105 divided by 98. Show the long division steps. | To divide 15105 by 98:
1. Determine how many times 98 goes into the leading digits of 15105.
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: 15105 / 98 = 154 with a remainder of 13.
Proof: (154 * 98) ... |
Add the fractions 3/12 and 13/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 11, which is 132.
2. Convert 3/12 by multiplying top and bottom by 11: 33/132.
3. Convert 13/11 by multiplying top and bottom by 12: 156/132.
4. Add the numerators: 33 + 156 = 189.
5. The resulting fraction is 189/132.
6. ... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^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: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
Compute the quotient and remainder of 35468 divided by 23. Show the long division steps. | To divide 35468 by 23:
1. Determine how many times 23 goes into the leading digits of 35468.
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: 35468 / 23 = 1542 with a remainder of 2.
Proof: (1542 * 23)... |
Compute the indefinite integral of f(x) = 4x^2. | To find the indefinite integral ∫ 4x^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: 4 / 3 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^2 dx = 4/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[2, -3]
B =
[2, -4]
[-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) = (-3*2) + (4*-1) = -10
C12 = (A11 * B12) + (A12 * B22) = (-3*-4) + (4*3) = 24
C21 = (A21 * B11) + (A22 * B21) = (2*2) + (-3*-1) = 7
C22 = (A21 * B12) + (A22 * B22) = (2*-4) + (-3*3) = -17
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 8, 1, 17, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 8 + 1 + 17 + 7) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.4, -2.6, -9.6, 6.4, -3.6]
Squared Differences: [88.36, 6.76, 92.16, 40.96, 12.96]
... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 10^2 = c^2
2. Calculate the squares: 289 + 100 = c^2
3. Sum the squares: 389 = c^2
4. Take the square root of both sides: c = √389
Final A... |
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 +... |
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 roots of the quadratic equation: 7x^2 + -8x + -2 = 0 | 1. Identify coefficients: a=7, b=-8, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -56 = 120.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √120) / 14.
5. Calculate roots: x1 = 1.3538893678645232, x2 = -0.21103222500738017.
Final... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 8, 5, 15, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 8 + 5 + 15 + 20) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, -4.8, -7.8, 2.2, 7.2]
Squared Differences: [10.24, 23.04, 60.84, 4.84, 51.84]
... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 15^2 = c^2
2. Calculate the squares: 144 + 225 = c^2
3. Sum the squares: 369 = c^2
4. Take the square root of both sides: c = √369
Final A... |
Multiply the following 2x2 matrices:
A =
[-5, 4]
[-1, -3]
B =
[-2, 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*-2) + (4*5) = 30
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (4*3) = 7
C21 = (A21 * B11) + (A22 * B21) = (-1*-2) + (-3*5) = -13
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (-3*3) = -10
Resu... |
Multiply the following 2x2 matrices:
A =
[4, -1]
[-3, 3]
B =
[-2, -1]
[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) = (4*-2) + (-1*5) = -13
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (-1*-5) = 1
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (3*5) = 21
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (3*-5) = -12
Re... |
Add the fractions 2/10 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 6, which is 30.
2. Convert 2/10 by multiplying top and bottom by 3: 6/30.
3. Convert 3/6 by multiplying top and bottom by 5: 15/30.
4. Add the numerators: 6 + 15 = 21.
5. The resulting fraction is 21/30.
6. Simplify by div... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 13. 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: 20^2 + 13^2 = c^2
2. Calculate the squares: 400 + 169 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 17, 16, 6, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 17 + 16 + 6 + 17) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 4.6, 3.6, -6.4, 4.6]
Squared Differences: [40.96, 21.16, 12.96, 40.96, 21.16]... |
Find the roots of the quadratic equation: 2x^2 + -7x + 4 = 0 | 1. Identify coefficients: a=2, b=-7, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 32 = 17.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √17) / 4.
5. Calculate roots: x1 = 2.7807764064044154, x2 = 0.7192235935955849.
Final Answer... |
Multiply the following 2x2 matrices:
A =
[2, 1]
[-5, 3]
B =
[2, 5]
[0, -1] | 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) + (1*0) = 4
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (1*-1) = 9
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (3*0) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*5) + (3*-1) = -28
Result Ma... |
Multiply the following 2x2 matrices:
A =
[5, 0]
[-5, -5]
B =
[-5, -5]
[-3, -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) + (0*-3) = -25
C12 = (A11 * B12) + (A12 * B22) = (5*-5) + (0*-2) = -25
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (-5*-3) = 40
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-5*-2) = 35
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.