prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
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 5/4 and 12/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 12, which is 12.
2. Convert 5/4 by multiplying top and bottom by 3: 15/12.
3. Convert 12/12 by multiplying top and bottom by 1: 12/12.
4. Add the numerators: 15 + 12 = 27.
5. The resulting fraction is 27/12.
6. Simplify by ... |
Compute the quotient and remainder of 5233 divided by 59. Show the long division steps. | To divide 5233 by 59:
1. Determine how many times 59 goes into the leading digits of 5233.
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: 5233 / 59 = 88 with a remainder of 41.
Proof: (88 * 59) + 41 ... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 19^2 = c^2
2. Calculate the squares: 225 + 361 = c^2
3. Sum the squares: 586 = c^2
4. Take the square root of both sides: c = √586
Final A... |
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 ... |
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 ... |
Multiply the following 2x2 matrices:
A =
[3, -3]
[-1, -5]
B =
[-1, 0]
[2, 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*-1) + (-3*2) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (-3*3) = -9
C21 = (A21 * B11) + (A22 * B21) = (-1*-1) + (-5*2) = -9
C22 = (A21 * B12) + (A22 * B22) = (-1*0) + (-5*3) = -15
Resu... |
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 |
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 ... |
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 |
Compute the quotient and remainder of 10093 divided by 5. Show the long division steps. | To divide 10093 by 5:
1. Determine how many times 5 goes into the leading digits of 10093.
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: 10093 / 5 = 2018 with a remainder of 3.
Proof: (2018 * 5) + 3... |
Compute the quotient and remainder of 72937 divided by 17. Show the long division steps. | To divide 72937 by 17:
1. Determine how many times 17 goes into the leading digits of 72937.
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: 72937 / 17 = 4290 with a remainder of 7.
Proof: (4290 * 17)... |
Add the fractions 13/2 and 6/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 8, which is 8.
2. Convert 13/2 by multiplying top and bottom by 4: 52/8.
3. Convert 6/8 by multiplying top and bottom by 1: 6/8.
4. Add the numerators: 52 + 6 = 58.
5. The resulting fraction is 58/8.
6. Simplify by dividing... |
Multiply the following 2x2 matrices:
A =
[5, 4]
[-2, -3]
B =
[4, 2]
[2, -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*4) + (4*2) = 28
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (4*-5) = -10
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (-3*2) = -14
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (-3*-5) = 11
Resul... |
Add the fractions 13/9 and 9/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 7, which is 63.
2. Convert 13/9 by multiplying top and bottom by 7: 91/63.
3. Convert 9/7 by multiplying top and bottom by 9: 81/63.
4. Add the numerators: 91 + 81 = 172.
5. The resulting fraction is 172/63.
Final Answer: 1... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[2, 1]
B =
[0, -1]
[-2, 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) = (-1*0) + (5*-2) = -10
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (5*0) = 1
C21 = (A21 * B11) + (A22 * B21) = (2*0) + (1*-2) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*-1) + (1*0) = -2
Result ... |
Multiply the following 2x2 matrices:
A =
[2, 4]
[-3, 4]
B =
[-1, -1]
[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) = (2*-1) + (4*2) = 6
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (4*-2) = -10
C21 = (A21 * B11) + (A22 * B21) = (-3*-1) + (4*2) = 11
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (4*-2) = -5
Resul... |
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 |
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 |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) 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) |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 18^2 = c^2
2. Calculate the squares: 36 + 324 = c^2
3. Sum the squares: 360 = c^2
4. Take the square root of both sides: c = √360
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-4, -3]
[2, -1]
B =
[-5, 2]
[5, -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*-5) + (-3*5) = 5
C12 = (A11 * B12) + (A12 * B22) = (-4*2) + (-3*-2) = -2
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-1*5) = -15
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (-1*-2) = 6
Resu... |
Add the fractions 1/13 and 3/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 2, which is 26.
2. Convert 1/13 by multiplying top and bottom by 2: 2/26.
3. Convert 3/2 by multiplying top and bottom by 13: 39/26.
4. Add the numerators: 2 + 39 = 41.
5. The resulting fraction is 41/26.
Final Answer: 41/... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 4, 16, 2, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 4 + 16 + 2 + 20) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.6, -5.6, 6.4, -7.6, 10.4]
Squared Differences: [12.96, 31.36, 40.96, 57.76, 108.16... |
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^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 |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 3, 16, 10, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 3 + 16 + 10 + 11) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, -8.0, 5.0, -1.0, 0.0]
Squared Differences: [16.0, 64.0, 25.0, 1.0, 0.0]
Su... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 10, 19, 2, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 10 + 19 + 2 + 12) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.2, 0.8, 9.8, -7.2, 2.8]
Squared Differences: [38.44, 0.64, 96.04, 51.84, 7.84]
... |
Add the fractions 3/2 and 1/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 5, which is 10.
2. Convert 3/2 by multiplying top and bottom by 5: 15/10.
3. Convert 1/5 by multiplying top and bottom by 2: 2/10.
4. Add the numerators: 15 + 2 = 17.
5. The resulting fraction is 17/10.
Final Answer: 17/10 |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 15, 6, 9, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 15 + 6 + 9 + 7) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, 6.6, -2.4, 0.6, -1.4]
Squared Differences: [11.56, 43.56, 5.76, 0.36, 1.96]
S... |
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 |
Multiply the following 2x2 matrices:
A =
[1, 0]
[2, 4]
B =
[-2, -1]
[-2, 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*-2) + (0*-2) = -2
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (0*5) = -1
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (4*-2) = -12
C22 = (A21 * B12) + (A22 * B22) = (2*-1) + (4*5) = 18
Result... |
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: [20, 5, 20, 3, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 5 + 20 + 3 + 19) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.6, -8.4, 6.6, -10.4, 5.6]
Squared Differences: [43.56, 70.56, 43.56, 108.16, 31.3... |
Add the fractions 4/6 and 12/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 11, which is 66.
2. Convert 4/6 by multiplying top and bottom by 11: 44/66.
3. Convert 12/11 by multiplying top and bottom by 6: 72/66.
4. Add the numerators: 44 + 72 = 116.
5. The resulting fraction is 116/66.
6. Simplify ... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
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) |
Multiply the following 2x2 matrices:
A =
[1, 3]
[3, -2]
B =
[5, -2]
[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) = (1*5) + (3*5) = 20
C12 = (A11 * B12) + (A12 * B22) = (1*-2) + (3*0) = -2
C21 = (A21 * B11) + (A22 * B21) = (3*5) + (-2*5) = 5
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-2*0) = -6
Result Mat... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 14, 13, 13, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 14 + 13 + 13 + 12) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.6, 3.4, 2.4, 2.4, 1.4]
Squared Differences: [92.16, 11.56, 5.76, 5.76, 1.96]
... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Add the fractions 2/11 and 6/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 15, which is 165.
2. Convert 2/11 by multiplying top and bottom by 15: 30/165.
3. Convert 6/15 by multiplying top and bottom by 11: 66/165.
4. Add the numerators: 30 + 66 = 96.
5. The resulting fraction is 96/165.
6. Simpl... |
Add the fractions 14/9 and 1/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 13, which is 117.
2. Convert 14/9 by multiplying top and bottom by 13: 182/117.
3. Convert 1/13 by multiplying top and bottom by 9: 9/117.
4. Add the numerators: 182 + 9 = 191.
5. The resulting fraction is 191/117.
Final An... |
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 ... |
Compute the quotient and remainder of 12751 divided by 97. Show the long division steps. | To divide 12751 by 97:
1. Determine how many times 97 goes into the leading digits of 12751.
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: 12751 / 97 = 131 with a remainder of 44.
Proof: (131 * 97) ... |
Find the roots of the quadratic equation: 9x^2 + 8x + -7 = 0 | 1. Identify coefficients: a=9, b=8, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -252 = 316.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √316) / 18.
5. Calculate roots: x1 = 0.5431327130350654, x2 = -1.4320216019239542.
Final... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
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 ... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, 2]
[2, -3]
B =
[1, 0]
[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) = (-1*1) + (2*3) = 5
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (2*-5) = -10
C21 = (A21 * B11) + (A22 * B21) = (2*1) + (-3*3) = -7
C22 = (A21 * B12) + (A22 * B22) = (2*0) + (-3*-5) = 15
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 8^2 = c^2
2. Calculate the squares: 400 + 64 = c^2
3. Sum the squares: 464 = c^2
4. Take the square root of both sides: c = √464
Final Ans... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 5x^2 + 6x + -6 = 0 | 1. Identify coefficients: a=5, b=6, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -120 = 156.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √156) / 10.
5. Calculate roots: x1 = 0.6489995996796797, x2 = -1.8489995996796798.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 13^2 = c^2
2. Calculate the squares: 121 + 169 = c^2
3. Sum the squares: 290 = c^2
4. Take the square root of both sides: c = √290
Final A... |
Add the fractions 4/15 and 12/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 7, which is 105.
2. Convert 4/15 by multiplying top and bottom by 7: 28/105.
3. Convert 12/7 by multiplying top and bottom by 15: 180/105.
4. Add the numerators: 28 + 180 = 208.
5. The resulting fraction is 208/105.
Final ... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) 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) |
Add the fractions 9/8 and 13/6. 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 9/8 by multiplying top and bottom by 3: 27/24.
3. Convert 13/6 by multiplying top and bottom by 4: 52/24.
4. Add the numerators: 27 + 52 = 79.
5. The resulting fraction is 79/24.
Final Answer: 79/... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 8^2 = c^2
2. Calculate the squares: 49 + 64 = c^2
3. Sum the squares: 113 = c^2
4. Take the square root of both sides: c = √113
Final Answe... |
Find the roots of the quadratic equation: 3x^2 + -6x + -6 = 0 | 1. Identify coefficients: a=3, b=-6, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -72 = 108.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √108) / 6.
5. Calculate roots: x1 = 2.732050807568877, x2 = -0.7320508075688773.
Final An... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 10, 2, 3, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 10 + 2 + 3 + 14) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.2, 1.2, -6.8, -5.8, 5.2]
Squared Differences: [38.44, 1.44, 46.24, 33.64, 27.04]
... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 8^2 = c^2
2. Calculate the squares: 324 + 64 = c^2
3. Sum the squares: 388 = c^2
4. Take the square root of both sides: c = √388
Final Ans... |
Compute the quotient and remainder of 57328 divided by 83. Show the long division steps. | To divide 57328 by 83:
1. Determine how many times 83 goes into the leading digits of 57328.
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: 57328 / 83 = 690 with a remainder of 58.
Proof: (690 * 83) ... |
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 ... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
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: [11, 2, 5, 13, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 2 + 5 + 13 + 16) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.6, -7.4, -4.4, 3.6, 6.6]
Squared Differences: [2.56, 54.76, 19.36, 12.96, 43.56]
... |
Find the roots of the quadratic equation: 6x^2 + 0x + -7 = 0 | 1. Identify coefficients: a=6, b=0, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -168 = 168.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √168) / 12.
5. Calculate roots: x1 = 1.0801234497346435, x2 = -1.0801234497346435.
Final A... |
Compute the quotient and remainder of 63248 divided by 51. Show the long division steps. | To divide 63248 by 51:
1. Determine how many times 51 goes into the leading digits of 63248.
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: 63248 / 51 = 1240 with a remainder of 8.
Proof: (1240 * 51)... |
Find the roots of the quadratic equation: 7x^2 + -8x + -10 = 0 | 1. Identify coefficients: a=7, b=-8, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -280 = 344.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √344) / 14.
5. Calculate roots: x1 = 1.8962312136422435, x2 = -0.7533740707851005.
Fina... |
Compute the quotient and remainder of 32269 divided by 3. Show the long division steps. | To divide 32269 by 3:
1. Determine how many times 3 goes into the leading digits of 32269.
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: 32269 / 3 = 10756 with a remainder of 1.
Proof: (10756 * 3) +... |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[-4, 0]
B =
[-1, -1]
[-2, -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) = (-1*-1) + (-5*-2) = 11
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (-5*-1) = 6
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (0*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (0*-1) = 4
Re... |
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 |
Compute the quotient and remainder of 67937 divided by 58. Show the long division steps. | To divide 67937 by 58:
1. Determine how many times 58 goes into the leading digits of 67937.
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: 67937 / 58 = 1171 with a remainder of 19.
Proof: (1171 * 58... |
Find the roots of the quadratic equation: 3x^2 + -4x + -8 = 0 | 1. Identify coefficients: a=3, b=-4, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -96 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √112) / 6.
5. Calculate roots: x1 = 2.4305008740430605, x2 = -1.0971675407097272.
Final A... |
Compute the quotient and remainder of 10015 divided by 94. Show the long division steps. | To divide 10015 by 94:
1. Determine how many times 94 goes into the leading digits of 10015.
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: 10015 / 94 = 106 with a remainder of 51.
Proof: (106 * 94) ... |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 11. 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 + 11^2 = c^2
2. Calculate the squares: 16 + 121 = c^2
3. Sum the squares: 137 = c^2
4. Take the square root of both sides: c = √137
Final Ans... |
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 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 indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^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: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Add the fractions 4/10 and 9/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 12, which is 60.
2. Convert 4/10 by multiplying top and bottom by 6: 24/60.
3. Convert 9/12 by multiplying top and bottom by 5: 45/60.
4. Add the numerators: 24 + 45 = 69.
5. The resulting fraction is 69/60.
6. Simplify by... |
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 = 11. 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 + 11^2 = c^2
2. Calculate the squares: 81 + 121 = c^2
3. Sum the squares: 202 = c^2
4. Take the square root of both sides: c = √202
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 9^2 = c^2
2. Calculate the squares: 225 + 81 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (4x + 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 = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Multiply the following 2x2 matrices:
A =
[-5, 1]
[-5, 1]
B =
[-4, 3]
[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) = (-5*-4) + (1*0) = 20
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (1*1) = -14
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (1*0) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*3) + (1*1) = -14
Resul... |
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 |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(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) |
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 ... |
Find the roots of the quadratic equation: 3x^2 + 3x + -2 = 0 | 1. Identify coefficients: a=3, b=3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -24 = 33.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √33) / 6.
5. Calculate roots: x1 = 0.4574271077563381, x2 = -1.457427107756338.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[-1, 1]
B =
[4, -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) = (1*4) + (5*-5) = -21
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (5*1) = 4
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (1*-5) = -9
C22 = (A21 * B12) + (A22 * B22) = (-1*-1) + (1*1) = 2
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 11 and b = 12. 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: 11^2 + 12^2 = c^2
2. Calculate the squares: 121 + 144 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final A... |
Multiply the following 2x2 matrices:
A =
[4, 1]
[3, 1]
B =
[4, -1]
[2, 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*4) + (1*2) = 18
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (1*5) = 1
C21 = (A21 * B11) + (A22 * B21) = (3*4) + (1*2) = 14
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (1*5) = 2
Result Matrix... |
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 |
Add the fractions 9/12 and 2/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 6, which is 12.
2. Convert 9/12 by multiplying top and bottom by 1: 9/12.
3. Convert 2/6 by multiplying top and bottom by 2: 4/12.
4. Add the numerators: 9 + 4 = 13.
5. The resulting fraction is 13/12.
Final Answer: 13/12 |
Add the fractions 12/10 and 3/2. 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 12/10 by multiplying top and bottom by 1: 12/10.
3. Convert 3/2 by multiplying top and bottom by 5: 15/10.
4. Add the numerators: 12 + 15 = 27.
5. The resulting fraction is 27/10.
Final Answer: 2... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Find the roots of the quadratic equation: 7x^2 + 8x + 8 = 0 | 1. Identify coefficients: a=7, b=8, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(7)(8) = 64 - 224 = -160.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √160i) / 14.
Final Answer: x = -0.5714285714285714 ± 0.9035079029052513i |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.