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Compute the quotient and remainder of 71306 divided by 79. Show the long division steps. | To divide 71306 by 79:
1. Determine how many times 79 goes into the leading digits of 71306.
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: 71306 / 79 = 902 with a remainder of 48.
Proof: (902 * 79) ... |
Add the fractions 5/13 and 2/11. 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 5/13 by multiplying top and bottom by 11: 55/143.
3. Convert 2/11 by multiplying top and bottom by 13: 26/143.
4. Add the numerators: 55 + 26 = 81.
5. The resulting fraction is 81/143.
Final An... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Find the roots of the quadratic equation: 3x^2 + 5x + -7 = 0 | 1. Identify coefficients: a=3, b=5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -84 = 109.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √109) / 6.
5. Calculate roots: x1 = 0.9067177514850918, x2 = -2.5733844181517584.
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 2, 9, 12, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 2 + 9 + 12 + 2) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, -5.2, 1.8, 4.8, -5.2]
Squared Differences: [14.44, 27.04, 3.24, 23.04, 27.04]
... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 17, 12, 6, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 17 + 12 + 6 + 14) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, 5.6, 0.6, -5.4, 2.6]
Squared Differences: [11.56, 31.36, 0.36, 29.16, 6.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 20, 10, 12, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 20 + 10 + 12 + 13) / 5 = 75 / 5 = 15.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, 5.0, -5.0, -3.0, -2.0]
Squared Differences: [25.0, 25.0, 25.0, 9.0, 4.0]
... |
Add the fractions 12/3 and 11/9. 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 12/3 by multiplying top and bottom by 3: 36/9.
3. Convert 11/9 by multiplying top and bottom by 1: 11/9.
4. Add the numerators: 36 + 11 = 47.
5. The resulting fraction is 47/9.
Final Answer: 47/9 |
Compute the quotient and remainder of 56687 divided by 2. Show the long division steps. | To divide 56687 by 2:
1. Determine how many times 2 goes into the leading digits of 56687.
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: 56687 / 2 = 28343 with a remainder of 1.
Proof: (28343 * 2) +... |
Multiply the following 2x2 matrices:
A =
[5, -5]
[4, -3]
B =
[2, 4]
[-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) = (5*2) + (-5*-4) = 30
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (-5*-5) = 45
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (-3*-4) = 20
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (-3*-5) = 31
Resul... |
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 +... |
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 |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 3, 5, 3, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 3 + 5 + 3 + 15) / 5 = 33 / 5 = 6.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, -3.6, -1.6, -3.6, 8.4]
Squared Differences: [0.16, 12.96, 2.56, 12.96, 70.56]
... |
Compute the quotient and remainder of 63008 divided by 80. Show the long division steps. | To divide 63008 by 80:
1. Determine how many times 80 goes into the leading digits of 63008.
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: 63008 / 80 = 787 with a remainder of 48.
Proof: (787 * 80) ... |
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 |
Find the roots of the quadratic equation: 9x^2 + 6x + 1 = 0 | 1. Identify coefficients: a=9, b=6, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 36 = 0.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √0) / 18.
5. Calculate roots: x1 = -0.3333333333333333, x2 = -0.3333333333333333.
Final Answe... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 8, 1, 15, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 8 + 1 + 15 + 9) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, 0.4, -6.6, 7.4, 1.4]
Squared Differences: [6.76, 0.16, 43.56, 54.76, 1.96]
Su... |
Compute the indefinite integral of f(x) = 6x^2. | To find the indefinite integral ∫ 6x^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: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + 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) |
Compute the quotient and remainder of 95405 divided by 57. Show the long division steps. | To divide 95405 by 57:
1. Determine how many times 57 goes into the leading digits of 95405.
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: 95405 / 57 = 1673 with a remainder of 44.
Proof: (1673 * 57... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 5, 5, 17, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 5 + 5 + 17 + 16) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.6, -7.4, -7.4, 4.6, 3.6]
Squared Differences: [43.56, 54.76, 54.76, 21.16, 12.96]... |
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 =
[-4, 3]
[0, -1]
B =
[-2, -4]
[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) = (-4*-2) + (3*4) = 20
C12 = (A11 * B12) + (A12 * B22) = (-4*-4) + (3*-3) = 7
C21 = (A21 * B11) + (A22 * B21) = (0*-2) + (-1*4) = -4
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (-1*-3) = 3
Resul... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
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 roots of the quadratic equation: 5x^2 + 8x + 2 = 0 | 1. Identify coefficients: a=5, b=8, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - 40 = 24.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √24) / 10.
5. Calculate roots: x1 = -0.31010205144336445, x2 = -1.2898979485566355.
Final An... |
Add the fractions 2/6 and 13/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 2, which is 6.
2. Convert 2/6 by multiplying top and bottom by 1: 2/6.
3. Convert 13/2 by multiplying top and bottom by 3: 39/6.
4. Add the numerators: 2 + 39 = 41.
5. The resulting fraction is 41/6.
Final Answer: 41/6 |
Compute the quotient and remainder of 56977 divided by 91. Show the long division steps. | To divide 56977 by 91:
1. Determine how many times 91 goes into the leading digits of 56977.
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: 56977 / 91 = 626 with a remainder of 11.
Proof: (626 * 91) ... |
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) = (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 +... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[2, 2]
B =
[-1, 1]
[-1, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-1) + (-3*-1) = 6
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (-3*2) = -9
C21 = (A21 * B11) + (A22 * B21) = (2*-1) + (2*-1) = -4
C22 = (A21 * B12) + (A22 * B22) = (2*1) + (2*2) = 6
Result ... |
Add the fractions 6/15 and 12/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 15, which is 15.
2. Convert 6/15 by multiplying top and bottom by 1: 6/15.
3. Convert 12/15 by multiplying top and bottom by 1: 12/15.
4. Add the numerators: 6 + 12 = 18.
5. The resulting fraction is 18/15.
6. Simplify by ... |
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 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) |
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) |
Multiply the following 2x2 matrices:
A =
[0, 3]
[5, 2]
B =
[5, -3]
[-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) = (0*5) + (3*-3) = -9
C12 = (A11 * B12) + (A12 * B22) = (0*-3) + (3*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (2*-3) = 19
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (2*0) = -15
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 20^2 + 3^2 = c^2
2. Calculate the squares: 400 + 9 = c^2
3. Sum the squares: 409 = c^2
4. Take the square root of both sides: c = √409
Final Answ... |
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 ... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Evaluate 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) |
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 ... |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[-2, -4]
[3, 0]
B =
[3, 2]
[-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) = (-2*3) + (-4*-2) = 2
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (-4*3) = -16
C21 = (A21 * B11) + (A22 * B21) = (3*3) + (0*-2) = 9
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (0*3) = 6
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 17^2 = c^2
2. Calculate the squares: 25 + 289 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
Add the fractions 15/14 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 4, which is 28.
2. Convert 15/14 by multiplying top and bottom by 2: 30/28.
3. Convert 6/4 by multiplying top and bottom by 7: 42/28.
4. Add the numerators: 30 + 42 = 72.
5. The resulting fraction is 72/28.
6. Simplify by ... |
Add the fractions 9/12 and 2/8. 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 9/12 by multiplying top and bottom by 2: 18/24.
3. Convert 2/8 by multiplying top and bottom by 3: 6/24.
4. Add the numerators: 18 + 6 = 24.
5. The resulting fraction is 24/24.
6. Simplify by div... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 11373 divided by 60. Show the long division steps. | To divide 11373 by 60:
1. Determine how many times 60 goes into the leading digits of 11373.
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: 11373 / 60 = 189 with a remainder of 33.
Proof: (189 * 60) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 9, 16, 18, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 9 + 16 + 18 + 1) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, -1.0, 6.0, 8.0, -9.0]
Squared Differences: [16.0, 1.0, 36.0, 64.0, 81.0]
Su... |
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) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 8^2 = c^2
2. Calculate the squares: 25 + 64 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
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 |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[-2, -3]
B =
[5, 1]
[-1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*5) + (-3*-1) = -12
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (-3*3) = -12
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (-3*-1) = -7
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (-3*3) = -11
... |
Find the roots of the quadratic equation: 10x^2 + -1x + -4 = 0 | 1. Identify coefficients: a=10, b=-1, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -160 = 161.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √161) / 20.
5. Calculate roots: x1 = 0.684428877022476, x2 = -0.584428877022476.
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 = 15 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: 15^2 + 14^2 = c^2
2. Calculate the squares: 225 + 196 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Compute the quotient and remainder of 5119 divided by 11. Show the long division steps. | To divide 5119 by 11:
1. Determine how many times 11 goes into the leading digits of 5119.
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: 5119 / 11 = 465 with a remainder of 4.
Proof: (465 * 11) + 4 ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 3, 19, 12, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 3 + 19 + 12 + 2) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, -5.0, 11.0, 4.0, -6.0]
Squared Differences: [16.0, 25.0, 121.0, 16.0, 36.0]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 12, 15, 9, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 12 + 15 + 9 + 11) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, 0.8, 3.8, -2.2, -0.2]
Squared Differences: [4.84, 0.64, 14.44, 4.84, 0.04]
... |
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: 4x^2 + -2x + 3 = 0 | 1. Identify coefficients: a=4, b=-2, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(4)(3) = 4 - 48 = -44.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √44i) / 8.
Final Answer: x = 0.25 ± 0.82915619758885i |
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 +... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Multiply the following 2x2 matrices:
A =
[4, -4]
[-5, 5]
B =
[5, -3]
[-5, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*5) + (-4*-5) = 40
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (-4*4) = -28
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (5*-5) = -50
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (5*4) = 35
Res... |
Multiply the following 2x2 matrices:
A =
[4, 5]
[3, 5]
B =
[-2, -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) = (4*-2) + (5*-2) = -18
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (5*3) = -5
C21 = (A21 * B11) + (A22 * B21) = (3*-2) + (5*-2) = -16
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (5*3) = 0
Result... |
Add the fractions 4/4 and 8/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 4/4 by multiplying top and bottom by 9: 36/36.
3. Convert 8/9 by multiplying top and bottom by 4: 32/36.
4. Add the numerators: 36 + 32 = 68.
5. The resulting fraction is 68/36.
6. Simplify by div... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 14^2 = c^2
2. Calculate the squares: 144 + 196 = c^2
3. Sum the squares: 340 = c^2
4. Take the square root of both sides: c = √340
Final A... |
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 +... |
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 |
Add the fractions 1/9 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 3, which is 9.
2. Convert 1/9 by multiplying top and bottom by 1: 1/9.
3. Convert 14/3 by multiplying top and bottom by 3: 42/9.
4. Add the numerators: 1 + 42 = 43.
5. The resulting fraction is 43/9.
Final Answer: 43/9 |
Compute the quotient and remainder of 18102 divided by 47. Show the long division steps. | To divide 18102 by 47:
1. Determine how many times 47 goes into the leading digits of 18102.
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: 18102 / 47 = 385 with a remainder of 7.
Proof: (385 * 47) +... |
Add the fractions 13/6 and 2/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 4, which is 12.
2. Convert 13/6 by multiplying top and bottom by 2: 26/12.
3. Convert 2/4 by multiplying top and bottom by 3: 6/12.
4. Add the numerators: 26 + 6 = 32.
5. The resulting fraction is 32/12.
6. Simplify by divi... |
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_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 indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 18^2 = c^2
2. Calculate the squares: 64 + 324 = 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 79232 divided by 19. Show the long division steps. | To divide 79232 by 19:
1. Determine how many times 19 goes into the leading digits of 79232.
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: 79232 / 19 = 4170 with a remainder of 2.
Proof: (4170 * 19)... |
Add the fractions 13/10 and 7/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 9, which is 90.
2. Convert 13/10 by multiplying top and bottom by 9: 117/90.
3. Convert 7/9 by multiplying top and bottom by 10: 70/90.
4. Add the numerators: 117 + 70 = 187.
5. The resulting fraction is 187/90.
Final Answ... |
Compute the quotient and remainder of 55571 divided by 14. Show the long division steps. | To divide 55571 by 14:
1. Determine how many times 14 goes into the leading digits of 55571.
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: 55571 / 14 = 3969 with a remainder of 5.
Proof: (3969 * 14)... |
Compute the quotient and remainder of 33076 divided by 61. Show the long division steps. | To divide 33076 by 61:
1. Determine how many times 61 goes into the leading digits of 33076.
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: 33076 / 61 = 542 with a remainder of 14.
Proof: (542 * 61) ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 4^2 = c^2
2. Calculate the squares: 400 + 16 = c^2
3. Sum the squares: 416 = c^2
4. Take the square root of both sides: c = √416
Final Ans... |
Find the roots of the quadratic equation: 1x^2 + 5x + -2 = 0 | 1. Identify coefficients: a=1, b=5, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -8 = 33.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √33) / 2.
5. Calculate roots: x1 = 0.3722813232690143, x2 = -5.372281323269014.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[-5, 5]
[-5, -2]
B =
[-2, -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*-2) + (5*2) = 20
C12 = (A11 * B12) + (A12 * B22) = (-5*-5) + (5*3) = 40
C21 = (A21 * B11) + (A22 * B21) = (-5*-2) + (-2*2) = 6
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-2*3) = 19
Resu... |
Multiply the following 2x2 matrices:
A =
[0, -5]
[-3, -4]
B =
[3, -3]
[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) = (0*3) + (-5*2) = -10
C12 = (A11 * B12) + (A12 * B22) = (0*-3) + (-5*-3) = 15
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (-4*2) = -17
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (-4*-3) = 21
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 6, 19, 8, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 6 + 19 + 8 + 5) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.0, -4.0, 9.0, -2.0, -5.0]
Squared Differences: [4.0, 16.0, 81.0, 4.0, 25.0]
Sum... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + 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 ... |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^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: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 1, 18, 17, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 1 + 18 + 17 + 10) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, -9.2, 7.8, 6.8, -0.2]
Squared Differences: [27.04, 84.64, 60.84, 46.24, 0.04]... |
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