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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 +... |
Add the fractions 10/5 and 11/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 5, which is 5.
2. Convert 10/5 by multiplying top and bottom by 1: 10/5.
3. Convert 11/5 by multiplying top and bottom by 1: 11/5.
4. Add the numerators: 10 + 11 = 21.
5. The resulting fraction is 21/5.
Final Answer: 21/5 |
Compute the quotient and remainder of 85870 divided by 39. Show the long division steps. | To divide 85870 by 39:
1. Determine how many times 39 goes into the leading digits of 85870.
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: 85870 / 39 = 2201 with a remainder of 31.
Proof: (2201 * 39... |
Find the roots of the quadratic equation: 10x^2 + 10x + 10 = 0 | 1. Identify coefficients: a=10, b=10, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 10^2 - 4(10)(10) = 100 - 400 = -300.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-10 ± √300i) / 20.
Final Answer: x = -0.5 ± 0.8660254037844387i |
Multiply the following 2x2 matrices:
A =
[0, 1]
[2, 0]
B =
[3, 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) = (0*3) + (1*5) = 5
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (1*-2) = -2
C21 = (A21 * B11) + (A22 * B21) = (2*3) + (0*5) = 6
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (0*-2) = 4
Result Matrix ... |
Add the fractions 11/14 and 14/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 11/14 by multiplying top and bottom by 1: 11/14.
3. Convert 14/14 by multiplying top and bottom by 1: 14/14.
4. Add the numerators: 11 + 14 = 25.
5. The resulting fraction is 25/14.
Final Answer... |
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 |
Multiply the following 2x2 matrices:
A =
[-2, 5]
[-4, 2]
B =
[-2, -3]
[-2, 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*-2) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (5*4) = 26
C21 = (A21 * B11) + (A22 * B21) = (-4*-2) + (2*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (2*4) = 20
Resu... |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^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: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 18, 8, 14, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 18 + 8 + 14 + 2) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, 6.8, -3.2, 2.8, -9.2]
Squared Differences: [7.84, 46.24, 10.24, 7.84, 84.64]
... |
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 |
Multiply the following 2x2 matrices:
A =
[-2, -2]
[3, -1]
B =
[2, -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*2) + (-2*-2) = 0
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (-2*2) = -2
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-1*-2) = 8
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (-1*2) = -5
Resul... |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 20. 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: 3^2 + 20^2 = c^2
2. Calculate the squares: 9 + 400 = c^2
3. Sum the squares: 409 = c^2
4. Take the square root of both sides: c = √409
Final Answ... |
Multiply the following 2x2 matrices:
A =
[0, 5]
[3, -3]
B =
[1, 5]
[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*1) + (5*4) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (5*3) = 15
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-3*4) = -9
C22 = (A21 * B12) + (A22 * B22) = (3*5) + (-3*3) = 6
Result Matri... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 6, 7, 5, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 6 + 7 + 5 + 10) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.4, -1.6, -0.6, -2.6, 2.4]
Squared Differences: [5.76, 2.56, 0.36, 6.76, 5.76]
Su... |
Find the roots of the quadratic equation: 9x^2 + 3x + -5 = 0 | 1. Identify coefficients: a=9, b=3, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -180 = 189.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √189) / 18.
5. Calculate roots: x1 = 0.5970959491593066, x2 = -0.93042928249264.
Final An... |
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) = (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 ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 14^2 = c^2
2. Calculate the squares: 81 + 196 = c^2
3. Sum the squares: 277 = c^2
4. Take the square root of both sides: c = √277
Final Ans... |
Multiply the following 2x2 matrices:
A =
[1, -1]
[-5, 5]
B =
[0, 2]
[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*0) + (-1*4) = -4
C12 = (A11 * B12) + (A12 * B22) = (1*2) + (-1*5) = -3
C21 = (A21 * B11) + (A22 * B21) = (-5*0) + (5*4) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (5*5) = 15
Result Ma... |
Evaluate the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 9x^2 + 4x + -10 = 0 | 1. Identify coefficients: a=9, b=4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -360 = 376.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √376) / 18.
5. Calculate roots: x1 = 0.8550399683147398, x2 = -1.2994844127591842.
Fina... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 10^2 = c^2
2. Calculate the squares: 169 + 100 = c^2
3. Sum the squares: 269 = c^2
4. Take the square root of both sides: c = √269
Final A... |
Compute the quotient and remainder of 47877 divided by 89. Show the long division steps. | To divide 47877 by 89:
1. Determine how many times 89 goes into the leading digits of 47877.
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: 47877 / 89 = 537 with a remainder of 84.
Proof: (537 * 89) ... |
Compute the quotient and remainder of 89814 divided by 82. Show the long division steps. | To divide 89814 by 82:
1. Determine how many times 82 goes into the leading digits of 89814.
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: 89814 / 82 = 1095 with a remainder of 24.
Proof: (1095 * 82... |
Multiply the following 2x2 matrices:
A =
[3, 2]
[3, -2]
B =
[2, -4]
[0, 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) = (3*2) + (2*0) = 6
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (2*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-2*0) = 6
C22 = (A21 * B12) + (A22 * B22) = (3*-4) + (-2*0) = -12
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[5, 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) = (-3*-1) + (4*-2) = -5
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (4*-1) = -1
C21 = (A21 * B11) + (A22 * B21) = (5*-1) + (0*-2) = -5
C22 = (A21 * B12) + (A22 * B22) = (5*-1) + (0*-1) = -5
Res... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[4, 4]
B =
[-3, -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*-3) + (5*-2) = -7
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (5*-1) = -4
C21 = (A21 * B11) + (A22 * B21) = (4*-3) + (4*-2) = -20
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (4*-1) = -8
Re... |
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 |
Compute the quotient and remainder of 55929 divided by 49. Show the long division steps. | To divide 55929 by 49:
1. Determine how many times 49 goes into the leading digits of 55929.
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: 55929 / 49 = 1141 with a remainder of 20.
Proof: (1141 * 49... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 17, 18, 17, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 17 + 18 + 17 + 6) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 3.2, 4.2, 3.2, -7.8]
Squared Differences: [7.84, 10.24, 17.64, 10.24, 60.84]... |
Add the fractions 11/14 and 8/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 6, which is 42.
2. Convert 11/14 by multiplying top and bottom by 3: 33/42.
3. Convert 8/6 by multiplying top and bottom by 7: 56/42.
4. Add the numerators: 33 + 56 = 89.
5. The resulting fraction is 89/42.
Final Answer: 8... |
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 ... |
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 +... |
Compute the quotient and remainder of 49916 divided by 52. Show the long division steps. | To divide 49916 by 52:
1. Determine how many times 52 goes into the leading digits of 49916.
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: 49916 / 52 = 959 with a remainder of 48.
Proof: (959 * 52) ... |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[-2, -4]
B =
[-3, -3]
[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) = (-3*-3) + (-3*2) = 3
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (-3*0) = 9
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (-4*2) = -2
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-4*0) = 6
Resu... |
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 |
Find the derivative of f(x) = (4x + 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 = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^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: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 19, 2, 8, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 19 + 2 + 8 + 20) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.6, 8.4, -8.6, -2.6, 9.4]
Squared Differences: [43.56, 70.56, 73.96, 6.76, 88.36]
... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-1, -2]
B =
[-1, 1]
[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) = (-1*-1) + (3*1) = 4
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (3*5) = 14
C21 = (A21 * B11) + (A22 * B21) = (-1*-1) + (-2*1) = -1
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (-2*5) = -11
Resul... |
Add the fractions 11/4 and 8/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 5, which is 20.
2. Convert 11/4 by multiplying top and bottom by 5: 55/20.
3. Convert 8/5 by multiplying top and bottom by 4: 32/20.
4. Add the numerators: 55 + 32 = 87.
5. The resulting fraction is 87/20.
Final Answer: 87/... |
Multiply the following 2x2 matrices:
A =
[3, -1]
[-2, -1]
B =
[-5, -2]
[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) = (3*-5) + (-1*1) = -16
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (-1*2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-2*-5) + (-1*1) = 9
C22 = (A21 * B12) + (A22 * B22) = (-2*-2) + (-1*2) = 2
Resu... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 9, 20, 1, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 9 + 20 + 1 + 9) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -1.6, 9.4, -9.6, -1.6]
Squared Differences: [11.56, 2.56, 88.36, 92.16, 2.56]
... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 13^2 = c^2
2. Calculate the squares: 16 + 169 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Add the fractions 11/15 and 9/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 2, which is 30.
2. Convert 11/15 by multiplying top and bottom by 2: 22/30.
3. Convert 9/2 by multiplying top and bottom by 15: 135/30.
4. Add the numerators: 22 + 135 = 157.
5. The resulting fraction is 157/30.
Final Answ... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[0, -1]
B =
[-2, -5]
[0, -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*-2) + (1*0) = 6
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (1*-2) = 13
C21 = (A21 * B11) + (A22 * B21) = (0*-2) + (-1*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (-1*-2) = 2
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 17, 15, 7, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 17 + 15 + 7 + 4) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, 4.8, 2.8, -5.2, -8.2]
Squared Differences: [33.64, 23.04, 7.84, 27.04, 67.24]
... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 5^2 = c^2
2. Calculate the squares: 196 + 25 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final Ans... |
Compute the quotient and remainder of 43600 divided by 33. Show the long division steps. | To divide 43600 by 33:
1. Determine how many times 33 goes into the leading digits of 43600.
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: 43600 / 33 = 1321 with a remainder of 7.
Proof: (1321 * 33)... |
Compute the quotient and remainder of 78938 divided by 12. Show the long division steps. | To divide 78938 by 12:
1. Determine how many times 12 goes into the leading digits of 78938.
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: 78938 / 12 = 6578 with a remainder of 2.
Proof: (6578 * 12)... |
Multiply the following 2x2 matrices:
A =
[0, -1]
[-4, -4]
B =
[3, 5]
[-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) = (0*3) + (-1*-2) = 2
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (-1*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (-4*3) + (-4*-2) = -4
C22 = (A21 * B12) + (A22 * B22) = (-4*5) + (-4*1) = -24
Resul... |
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 |
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 roots of the quadratic equation: 8x^2 + 3x + 9 = 0 | 1. Identify coefficients: a=8, b=3, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(8)(9) = 9 - 288 = -279.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √279i) / 16.
Final Answer: x = -0.1875 ± 1.0439558180306292i |
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 |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 6. 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 + 6^2 = c^2
2. Calculate the squares: 36 + 36 = c^2
3. Sum the squares: 72 = c^2
4. Take the square root of both sides: c = √72
Final Answer:... |
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 90679 divided by 65. Show the long division steps. | To divide 90679 by 65:
1. Determine how many times 65 goes into the leading digits of 90679.
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: 90679 / 65 = 1395 with a remainder of 4.
Proof: (1395 * 65)... |
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 indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^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: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^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: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
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 ... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 8^2 = c^2
2. Calculate the squares: 256 + 64 = c^2
3. Sum the squares: 320 = c^2
4. Take the square root of both sides: c = √320
Final Ans... |
Compute the quotient and remainder of 41074 divided by 57. Show the long division steps. | To divide 41074 by 57:
1. Determine how many times 57 goes into the leading digits of 41074.
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: 41074 / 57 = 720 with a remainder of 34.
Proof: (720 * 57) ... |
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 ... |
Add the fractions 11/5 and 12/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 5, which is 5.
2. Convert 11/5 by multiplying top and bottom by 1: 11/5.
3. Convert 12/5 by multiplying top and bottom by 1: 12/5.
4. Add the numerators: 11 + 12 = 23.
5. The resulting fraction is 23/5.
Final Answer: 23/5 |
Find the derivative of f(x) = (2x + 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 = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
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 |
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 roots of the quadratic equation: 2x^2 + 0x + 5 = 0 | 1. Identify coefficients: a=2, b=0, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(2)(5) = 0 - 40 = -40.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √40i) / 4.
Final Answer: x = 0.0 ± 1.5811388300841898i |
Find the roots of the quadratic equation: 1x^2 + 3x + -5 = 0 | 1. Identify coefficients: a=1, b=3, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -20 = 29.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √29) / 2.
5. Calculate roots: x1 = 1.1925824035672519, x2 = -4.192582403567252.
Final Answe... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 5^2 = c^2
2. Calculate the squares: 400 + 25 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 15^2 = c^2
2. Calculate the squares: 256 + 225 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final A... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 9^2 = c^2
2. Calculate the squares: 9 + 81 = c^2
3. Sum the squares: 90 = c^2
4. Take the square root of both sides: c = √90
Final Answer: ... |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[3, 5]
[-5, 1]
B =
[-5, 3]
[-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) = (3*-5) + (5*-2) = -25
C12 = (A11 * B12) + (A12 * B22) = (3*3) + (5*-1) = 4
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (1*-2) = 23
C22 = (A21 * B12) + (A22 * B22) = (-5*3) + (1*-1) = -16
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 12, 11, 11, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 12 + 11 + 11 + 5) / 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, 2.0, 1.0, 1.0, -5.0]
Squared Differences: [1.0, 4.0, 1.0, 1.0, 25.0]
Sum o... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 76915 divided by 30. Show the long division steps. | To divide 76915 by 30:
1. Determine how many times 30 goes into the leading digits of 76915.
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: 76915 / 30 = 2563 with a remainder of 25.
Proof: (2563 * 30... |
Find the roots of the quadratic equation: 2x^2 + 10x + -9 = 0 | 1. Identify coefficients: a=2, b=10, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -72 = 172.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √172) / 4.
5. Calculate roots: x1 = 0.7787192621510002, x2 = -5.778719262151.
Final An... |
Compute the quotient and remainder of 87101 divided by 76. Show the long division steps. | To divide 87101 by 76:
1. Determine how many times 76 goes into the leading digits of 87101.
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: 87101 / 76 = 1146 with a remainder of 5.
Proof: (1146 * 76)... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 1, 4, 4, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 1 + 4 + 4 + 4) / 5 = 19 / 5 = 3.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.2, -2.8, 0.2, 0.2, 0.2]
Squared Differences: [4.84, 7.84, 0.04, 0.04, 0.04]
Sum of... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[-2, 3]
B =
[-2, -5]
[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*-2) + (-1*1) = 7
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (-1*-2) = 22
C21 = (A21 * B11) + (A22 * B21) = (-2*-2) + (3*1) = 7
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (3*-2) = 4
Resu... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^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: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Find the roots of the quadratic equation: 10x^2 + -5x + 0 = 0 | 1. Identify coefficients: a=10, b=-5, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 0 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √25) / 20.
5. Calculate roots: x1 = 0.5, x2 = 0.0.
Final Answer: x = 0.5, 0.0 |
Find the roots of the quadratic equation: 10x^2 + 1x + -7 = 0 | 1. Identify coefficients: a=10, b=1, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -280 = 281.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √281) / 20.
5. Calculate roots: x1 = 0.7881527307120105, x2 = -0.8881527307120105.
Final... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 20, 1, 18, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 20 + 1 + 18 + 13) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.6, 8.4, -10.6, 6.4, 1.4]
Squared Differences: [31.36, 70.56, 112.36, 40.96, 1.96... |
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 60342 divided by 7. Show the long division steps. | To divide 60342 by 7:
1. Determine how many times 7 goes into the leading digits of 60342.
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: 60342 / 7 = 8620 with a remainder of 2.
Proof: (8620 * 7) + 2... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 10, 4, 8, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 10 + 4 + 8 + 8) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, 1.8, -4.2, -0.2, -0.2]
Squared Differences: [7.84, 3.24, 17.64, 0.04, 0.04]
S... |
Add the fractions 12/12 and 15/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 8, which is 24.
2. Convert 12/12 by multiplying top and bottom by 2: 24/24.
3. Convert 15/8 by multiplying top and bottom by 3: 45/24.
4. Add the numerators: 24 + 45 = 69.
5. The resulting fraction is 69/24.
6. Simplify by... |
Find the roots of the quadratic equation: 9x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=9, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -36 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √100) / 18.
5. Calculate roots: x1 = 1.0, x2 = -0.1111111111111111.
Final Answer: x = 1.0... |
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