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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 ... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 18, 11, 14, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 18 + 11 + 14 + 6) / 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, 8.0, 1.0, 4.0, -4.0]
Squared Differences: [81.0, 64.0, 1.0, 16.0, 16.0]
Su... |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[0, -4]
B =
[-5, -5]
[-3, 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*-5) + (5*-3) = 0
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (5*0) = 15
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (-4*-3) = 12
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (-4*0) = 0
Resul... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 4^2 = c^2
2. Calculate the squares: 16 + 16 = c^2
3. Sum the squares: 32 = c^2
4. Take the square root of both sides: c = √32
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[5, -3]
[5, 3]
B =
[0, -4]
[-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) = (5*0) + (-3*-1) = 3
C12 = (A11 * B12) + (A12 * B22) = (5*-4) + (-3*2) = -26
C21 = (A21 * B11) + (A22 * B21) = (5*0) + (3*-1) = -3
C22 = (A21 * B12) + (A22 * B22) = (5*-4) + (3*2) = -14
Result... |
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) = 2x^1. | To find the indefinite integral ∫ 2x^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: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^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 quotient and remainder of 40442 divided by 81. Show the long division steps. | To divide 40442 by 81:
1. Determine how many times 81 goes into the leading digits of 40442.
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: 40442 / 81 = 499 with a remainder of 23.
Proof: (499 * 81) ... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) 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) |
Multiply the following 2x2 matrices:
A =
[0, -1]
[1, 1]
B =
[-4, 5]
[-3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*-4) + (-1*-3) = 3
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (-1*-4) = 4
C21 = (A21 * B11) + (A22 * B21) = (1*-4) + (1*-3) = -7
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (1*-4) = 1
Result M... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[1, 2]
B =
[4, -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) = (0*4) + (-4*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (-4*-5) = 20
C21 = (A21 * B11) + (A22 * B21) = (1*4) + (2*-1) = 2
C22 = (A21 * B12) + (A22 * B22) = (1*-1) + (2*-5) = -11
Result... |
Add the fractions 14/3 and 14/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 9, which is 9.
2. Convert 14/3 by multiplying top and bottom by 3: 42/9.
3. Convert 14/9 by multiplying top and bottom by 1: 14/9.
4. Add the numerators: 42 + 14 = 56.
5. The resulting fraction is 56/9.
Final Answer: 56/9 |
Find the roots of the quadratic equation: 4x^2 + -7x + 7 = 0 | 1. Identify coefficients: a=4, b=-7, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(4)(7) = 49 - 112 = -63.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √63i) / 8.
Final Answer: x = 0.875 ± 0.9921567416492215i |
Compute the quotient and remainder of 91447 divided by 24. Show the long division steps. | To divide 91447 by 24:
1. Determine how many times 24 goes into the leading digits of 91447.
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: 91447 / 24 = 3810 with a remainder of 7.
Proof: (3810 * 24)... |
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) |
Add the fractions 11/2 and 14/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 10, which is 10.
2. Convert 11/2 by multiplying top and bottom by 5: 55/10.
3. Convert 14/10 by multiplying top and bottom by 1: 14/10.
4. Add the numerators: 55 + 14 = 69.
5. The resulting fraction is 69/10.
Final Answer: ... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 7, 16, 1, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 7 + 16 + 1 + 9) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, 0.2, 9.2, -5.8, 2.2]
Squared Differences: [33.64, 0.04, 84.64, 33.64, 4.84]
S... |
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:... |
Find the roots of the quadratic equation: 5x^2 + 10x + -6 = 0 | 1. Identify coefficients: a=5, b=10, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -120 = 220.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √220) / 10.
5. Calculate roots: x1 = 0.4832396974191326, x2 = -2.4832396974191324.
Fi... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 2, 17, 11, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 2 + 17 + 11 + 1) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, -6.4, 8.6, 2.6, -7.4]
Squared Differences: [6.76, 40.96, 73.96, 6.76, 54.76]
... |
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 ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 7^2 = c^2
2. Calculate the squares: 196 + 49 = c^2
3. Sum the squares: 245 = c^2
4. Take the square root of both sides: c = √245
Final Ans... |
Add the fractions 12/11 and 9/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 11, which is 11.
2. Convert 12/11 by multiplying top and bottom by 1: 12/11.
3. Convert 9/11 by multiplying top and bottom by 1: 9/11.
4. Add the numerators: 12 + 9 = 21.
5. The resulting fraction is 21/11.
Final Answer: 2... |
Find the roots of the quadratic equation: 3x^2 + -5x + -8 = 0 | 1. Identify coefficients: a=3, b=-5, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -96 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √121) / 6.
5. Calculate roots: x1 = 2.6666666666666665, x2 = -1.0.
Final Answer: x = 2.66... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 13, 16, 6, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 13 + 16 + 6 + 15) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, -0.4, 2.6, -7.4, 1.6]
Squared Differences: [12.96, 0.16, 6.76, 54.76, 2.56]
... |
Compute the quotient and remainder of 87490 divided by 61. Show the long division steps. | To divide 87490 by 61:
1. Determine how many times 61 goes into the leading digits of 87490.
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: 87490 / 61 = 1434 with a remainder of 16.
Proof: (1434 * 61... |
Find the roots of the quadratic equation: 7x^2 + -7x + 1 = 0 | 1. Identify coefficients: a=7, b=-7, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 28 = 21.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √21) / 14.
5. Calculate roots: x1 = 0.8273268353539885, x2 = 0.17267316464601143.
Final Answ... |
Add the fractions 3/8 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 13, which is 104.
2. Convert 3/8 by multiplying top and bottom by 13: 39/104.
3. Convert 6/13 by multiplying top and bottom by 8: 48/104.
4. Add the numerators: 39 + 48 = 87.
5. The resulting fraction is 87/104.
Final Answe... |
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 |
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 quotient and remainder of 45924 divided by 95. Show the long division steps. | To divide 45924 by 95:
1. Determine how many times 95 goes into the leading digits of 45924.
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: 45924 / 95 = 483 with a remainder of 39.
Proof: (483 * 95) ... |
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 ... |
Multiply the following 2x2 matrices:
A =
[-5, 3]
[2, -5]
B =
[3, -5]
[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) = (-5*3) + (3*2) = -9
C12 = (A11 * B12) + (A12 * B22) = (-5*-5) + (3*3) = 34
C21 = (A21 * B11) + (A22 * B21) = (2*3) + (-5*2) = -4
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-5*3) = -25
Result... |
Find the roots of the quadratic equation: 7x^2 + 9x + -7 = 0 | 1. Identify coefficients: a=7, b=9, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -196 = 277.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √277) / 14.
5. Calculate roots: x1 = 0.5459512126495171, x2 = -1.8316654983638028.
Final... |
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 |
Compute the quotient and remainder of 2635 divided by 60. Show the long division steps. | To divide 2635 by 60:
1. Determine how many times 60 goes into the leading digits of 2635.
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: 2635 / 60 = 43 with a remainder of 55.
Proof: (43 * 60) + 55 ... |
Compute the quotient and remainder of 19474 divided by 76. Show the long division steps. | To divide 19474 by 76:
1. Determine how many times 76 goes into the leading digits of 19474.
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: 19474 / 76 = 256 with a remainder of 18.
Proof: (256 * 76) ... |
Compute the quotient and remainder of 48130 divided by 54. Show the long division steps. | To divide 48130 by 54:
1. Determine how many times 54 goes into the leading digits of 48130.
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: 48130 / 54 = 891 with a remainder of 16.
Proof: (891 * 54) ... |
Multiply the following 2x2 matrices:
A =
[-3, -5]
[0, -3]
B =
[3, -1]
[-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) = (-3*3) + (-5*-3) = 6
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (-5*2) = -7
C21 = (A21 * B11) + (A22 * B21) = (0*3) + (-3*-3) = 9
C22 = (A21 * B12) + (A22 * B22) = (0*-1) + (-3*2) = -6
Resul... |
Multiply the following 2x2 matrices:
A =
[3, -1]
[-4, 5]
B =
[0, -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*0) + (-1*-2) = 2
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (-1*0) = -9
C21 = (A21 * B11) + (A22 * B21) = (-4*0) + (5*-2) = -10
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (5*0) = 12
Resul... |
Compute the quotient and remainder of 24642 divided by 36. Show the long division steps. | To divide 24642 by 36:
1. Determine how many times 36 goes into the leading digits of 24642.
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: 24642 / 36 = 684 with a remainder of 18.
Proof: (684 * 36) ... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^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: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^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: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Compute the quotient and remainder of 5705 divided by 62. Show the long division steps. | To divide 5705 by 62:
1. Determine how many times 62 goes into the leading digits of 5705.
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: 5705 / 62 = 92 with a remainder of 1.
Proof: (92 * 62) + 1 = ... |
Multiply the following 2x2 matrices:
A =
[2, 2]
[0, -3]
B =
[-1, -3]
[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) = (2*-1) + (2*5) = 8
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (2*-3) = -12
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (-3*5) = -15
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (-3*-3) = 9
Resul... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[2, 5]
B =
[3, -2]
[-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*3) + (5*-2) = -13
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (5*0) = 2
C21 = (A21 * B11) + (A22 * B21) = (2*3) + (5*-2) = -4
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (5*0) = -4
Result ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 17, 15, 12, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 17 + 15 + 12 + 12) / 5 = 74 / 5 = 14.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, 2.2, 0.2, -2.8, -2.8]
Squared Differences: [10.24, 4.84, 0.04, 7.84, 7.84]
... |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^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: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + 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) |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 16. 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 + 16^2 = c^2
2. Calculate the squares: 144 + 256 = c^2
3. Sum the squares: 400 = c^2
4. Take the square root of both sides: c = √400
Final A... |
Find the roots of the quadratic equation: 8x^2 + 2x + 9 = 0 | 1. Identify coefficients: a=8, b=2, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(8)(9) = 4 - 288 = -284.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √284i) / 16.
Final Answer: x = -0.125 ± 1.0532687216470449i |
Find the roots of the quadratic equation: 7x^2 + -9x + -8 = 0 | 1. Identify coefficients: a=7, b=-9, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -224 = 305.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √305) / 14.
5. Calculate roots: x1 = 1.890303514040927, x2 = -0.6045892283266413.
Final ... |
Add the fractions 12/8 and 5/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 12/8 by multiplying top and bottom by 1: 12/8.
3. Convert 5/8 by multiplying top and bottom by 1: 5/8.
4. Add the numerators: 12 + 5 = 17.
5. The resulting fraction is 17/8.
Final Answer: 17/8 |
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 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 |
Multiply the following 2x2 matrices:
A =
[1, 3]
[-4, 3]
B =
[0, -4]
[-4, -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*0) + (3*-4) = -12
C12 = (A11 * B12) + (A12 * B22) = (1*-4) + (3*-4) = -16
C21 = (A21 * B11) + (A22 * B21) = (-4*0) + (3*-4) = -12
C22 = (A21 * B12) + (A22 * B22) = (-4*-4) + (3*-4) = 4
Res... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 7, 10, 19, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 7 + 10 + 19 + 5) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, -2.8, 0.2, 9.2, -4.8]
Squared Differences: [3.24, 7.84, 0.04, 84.64, 23.04]
... |
Find the roots of the quadratic equation: 4x^2 + 3x + 10 = 0 | 1. Identify coefficients: a=4, b=3, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(4)(10) = 9 - 160 = -151.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √151i) / 8.
Final Answer: x = -0.375 ± 1.5360257159305635i |
Add the fractions 10/11 and 15/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 6, which is 66.
2. Convert 10/11 by multiplying top and bottom by 6: 60/66.
3. Convert 15/6 by multiplying top and bottom by 11: 165/66.
4. Add the numerators: 60 + 165 = 225.
5. The resulting fraction is 225/66.
6. Simpli... |
Compute the quotient and remainder of 33175 divided by 58. Show the long division steps. | To divide 33175 by 58:
1. Determine how many times 58 goes into the leading digits of 33175.
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: 33175 / 58 = 571 with a remainder of 57.
Proof: (571 * 58) ... |
Add the fractions 9/4 and 15/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 2, which is 4.
2. Convert 9/4 by multiplying top and bottom by 1: 9/4.
3. Convert 15/2 by multiplying top and bottom by 2: 30/4.
4. Add the numerators: 9 + 30 = 39.
5. The resulting fraction is 39/4.
Final Answer: 39/4 |
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 ... |
Add the fractions 8/7 and 15/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 8, which is 56.
2. Convert 8/7 by multiplying top and bottom by 8: 64/56.
3. Convert 15/8 by multiplying top and bottom by 7: 105/56.
4. Add the numerators: 64 + 105 = 169.
5. The resulting fraction is 169/56.
Final Answer:... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 17, 2, 3, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 17 + 2 + 3 + 12) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.6, 6.6, -8.4, -7.4, 1.6]
Squared Differences: [57.76, 43.56, 70.56, 54.76, 2.56]
... |
Add the fractions 10/13 and 5/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 11, which is 143.
2. Convert 10/13 by multiplying top and bottom by 11: 110/143.
3. Convert 5/11 by multiplying top and bottom by 13: 65/143.
4. Add the numerators: 110 + 65 = 175.
5. The resulting fraction is 175/143.
Fin... |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + 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 +... |
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 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 +... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) 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 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 |
Find the roots of the quadratic equation: 4x^2 + 1x + 6 = 0 | 1. Identify coefficients: a=4, b=1, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(4)(6) = 1 - 96 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √95i) / 8.
Final Answer: x = -0.125 ± 1.2183492931011204i |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 70622 divided by 17. Show the long division steps. | To divide 70622 by 17:
1. Determine how many times 17 goes into the leading digits of 70622.
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: 70622 / 17 = 4154 with a remainder of 4.
Proof: (4154 * 17)... |
Add the fractions 5/7 and 12/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 3, which is 21.
2. Convert 5/7 by multiplying top and bottom by 3: 15/21.
3. Convert 12/3 by multiplying top and bottom by 7: 84/21.
4. Add the numerators: 15 + 84 = 99.
5. The resulting fraction is 99/21.
6. Simplify by di... |
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 |
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(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 |
Add the fractions 1/14 and 8/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 15, which is 210.
2. Convert 1/14 by multiplying top and bottom by 15: 15/210.
3. Convert 8/15 by multiplying top and bottom by 14: 112/210.
4. Add the numerators: 15 + 112 = 127.
5. The resulting fraction is 127/210.
Fina... |
Multiply the following 2x2 matrices:
A =
[4, 3]
[2, -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) + (3*-5) = -23
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (3*5) = 11
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (-3*-5) = 11
C22 = (A21 * B12) + (A22 * B22) = (2*-1) + (-3*5) = -17
Res... |
Add the fractions 3/9 and 10/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 2, which is 18.
2. Convert 3/9 by multiplying top and bottom by 2: 6/18.
3. Convert 10/2 by multiplying top and bottom by 9: 90/18.
4. Add the numerators: 6 + 90 = 96.
5. The resulting fraction is 96/18.
6. Simplify by divi... |
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: [9, 7, 14, 14, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 7 + 14 + 14 + 13) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, -4.4, 2.6, 2.6, 1.6]
Squared Differences: [5.76, 19.36, 6.76, 6.76, 2.56]
... |
Find the roots of the quadratic equation: 3x^2 + -6x + 0 = 0 | 1. Identify coefficients: a=3, b=-6, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 0 = 36.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √36) / 6.
5. Calculate roots: x1 = 2.0, x2 = 0.0.
Final Answer: x = 2.0, 0.0 |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 20, 11, 1, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 20 + 11 + 1 + 9) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.0, 11.0, 2.0, -8.0, 0.0]
Squared Differences: [25.0, 121.0, 4.0, 64.0, 0.0]
Sum... |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[-2, 0]
B =
[2, -4]
[-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) = (-4*2) + (-2*-3) = -2
C12 = (A11 * B12) + (A12 * B22) = (-4*-4) + (-2*-5) = 26
C21 = (A21 * B11) + (A22 * B21) = (-2*2) + (0*-3) = -4
C22 = (A21 * B12) + (A22 * B22) = (-2*-4) + (0*-5) = 8
Re... |
Add the fractions 13/9 and 9/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 4, which is 36.
2. Convert 13/9 by multiplying top and bottom by 4: 52/36.
3. Convert 9/4 by multiplying top and bottom by 9: 81/36.
4. Add the numerators: 52 + 81 = 133.
5. The resulting fraction is 133/36.
Final Answer: 1... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 8^2 = c^2
2. Calculate the squares: 361 + 64 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
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 + -3x + -3 = 0 | 1. Identify coefficients: a=10, b=-3, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -120 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √129) / 20.
5. Calculate roots: x1 = 0.7178908345800273, x2 = -0.41789083458002735.
Fina... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 11^2 = c^2
2. Calculate the squares: 121 + 121 = c^2
3. Sum the squares: 242 = c^2
4. Take the square root of both sides: c = √242
Final A... |
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 ... |
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