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Find the roots of the quadratic equation: 8x^2 + 10x + 2 = 0 | 1. Identify coefficients: a=8, b=10, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 64 = 36.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √36) / 16.
5. Calculate roots: x1 = -0.25, x2 = -1.0.
Final Answer: x = -0.25, -1.0 |
Find the roots of the quadratic equation: 5x^2 + -7x + 0 = 0 | 1. Identify coefficients: a=5, b=-7, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 0 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √49) / 10.
5. Calculate roots: x1 = 1.4, x2 = 0.0.
Final Answer: x = 1.4, 0.0 |
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 |
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) = 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 |
Compute the quotient and remainder of 86428 divided by 76. Show the long division steps. | To divide 86428 by 76:
1. Determine how many times 76 goes into the leading digits of 86428.
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: 86428 / 76 = 1137 with a remainder of 16.
Proof: (1137 * 76... |
Multiply the following 2x2 matrices:
A =
[2, 2]
[-2, 4]
B =
[0, 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) = (2*0) + (2*4) = 8
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (2*-3) = 4
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (4*4) = 16
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (4*-3) = -22
Result Mat... |
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 |
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 |
Add the fractions 10/9 and 12/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 9, which is 9.
2. Convert 10/9 by multiplying top and bottom by 1: 10/9.
3. Convert 12/9 by multiplying top and bottom by 1: 12/9.
4. Add the numerators: 10 + 12 = 22.
5. The resulting fraction is 22/9.
Final Answer: 22/9 |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
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 +... |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 10^2 = c^2
2. Calculate the squares: 400 + 100 = c^2
3. Sum the squares: 500 = c^2
4. Take the square root of both sides: c = √500
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 18, 8, 18, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 18 + 8 + 18 + 14) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, 3.8, -6.2, 3.8, -0.2]
Squared Differences: [1.44, 14.44, 38.44, 14.44, 0.04]... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 3, 8, 18, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 3 + 8 + 18 + 14) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, -7.8, -2.8, 7.2, 3.2]
Squared Differences: [0.04, 60.84, 7.84, 51.84, 10.24]
... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Add the fractions 8/2 and 9/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 7, which is 14.
2. Convert 8/2 by multiplying top and bottom by 7: 56/14.
3. Convert 9/7 by multiplying top and bottom by 2: 18/14.
4. Add the numerators: 56 + 18 = 74.
5. The resulting fraction is 74/14.
6. Simplify by div... |
Compute the quotient and remainder of 68721 divided by 11. Show the long division steps. | To divide 68721 by 11:
1. Determine how many times 11 goes into the leading digits of 68721.
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: 68721 / 11 = 6247 with a remainder of 4.
Proof: (6247 * 11)... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[2, -3]
B =
[-4, 2]
[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*-4) + (-2*2) = -8
C12 = (A11 * B12) + (A12 * B22) = (1*2) + (-2*-1) = 4
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (-3*2) = -14
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (-3*-1) = 7
Result... |
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 ... |
Add the fractions 8/10 and 5/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 3, which is 30.
2. Convert 8/10 by multiplying top and bottom by 3: 24/30.
3. Convert 5/3 by multiplying top and bottom by 10: 50/30.
4. Add the numerators: 24 + 50 = 74.
5. The resulting fraction is 74/30.
6. Simplify by ... |
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) |
Add the fractions 8/11 and 10/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 8, which is 88.
2. Convert 8/11 by multiplying top and bottom by 8: 64/88.
3. Convert 10/8 by multiplying top and bottom by 11: 110/88.
4. Add the numerators: 64 + 110 = 174.
5. The resulting fraction is 174/88.
6. Simplif... |
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 |
Find the roots of the quadratic equation: 2x^2 + -1x + -5 = 0 | 1. Identify coefficients: a=2, b=-1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -40 = 41.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √41) / 4.
5. Calculate roots: x1 = 1.8507810593582121, x2 = -1.3507810593582121.
Final Answ... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[2, -4]
B =
[-3, 1]
[1, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-3) + (0*1) = 15
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (0*4) = -5
C21 = (A21 * B11) + (A22 * B21) = (2*-3) + (-4*1) = -10
C22 = (A21 * B12) + (A22 * B22) = (2*1) + (-4*4) = -14
Resul... |
Compute the indefinite integral of f(x) = 4x^3. | To find the indefinite integral ∫ 4x^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: 4 / 4 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^3 dx = 1x^4 + C |
Multiply the following 2x2 matrices:
A =
[0, -3]
[3, 5]
B =
[-5, -1]
[-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) = (0*-5) + (-3*-5) = 15
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (-3*4) = -12
C21 = (A21 * B11) + (A22 * B21) = (3*-5) + (5*-5) = -40
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (5*4) = 17
Res... |
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: 1x^2 + -3x + 1 = 0 | 1. Identify coefficients: a=1, b=-3, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - 4 = 5.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √5) / 2.
5. Calculate roots: x1 = 2.618033988749895, x2 = 0.3819660112501051.
Final Answer: x =... |
Compute the quotient and remainder of 65064 divided by 82. Show the long division steps. | To divide 65064 by 82:
1. Determine how many times 82 goes into the leading digits of 65064.
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: 65064 / 82 = 793 with a remainder of 38.
Proof: (793 * 82) ... |
Add the fractions 12/7 and 13/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 14, which is 14.
2. Convert 12/7 by multiplying top and bottom by 2: 24/14.
3. Convert 13/14 by multiplying top and bottom by 1: 13/14.
4. Add the numerators: 24 + 13 = 37.
5. The resulting fraction is 37/14.
Final Answer: ... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 14^2 = c^2
2. Calculate the squares: 289 + 196 = c^2
3. Sum the squares: 485 = c^2
4. Take the square root of both sides: c = √485
Final A... |
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) = 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 |
Find the roots of the quadratic equation: 1x^2 + -9x + -5 = 0 | 1. Identify coefficients: a=1, b=-9, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -20 = 101.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √101) / 2.
5. Calculate roots: x1 = 9.524937810560445, x2 = -0.524937810560445.
Final Ans... |
Find the roots of the quadratic equation: 1x^2 + 8x + -3 = 0 | 1. Identify coefficients: a=1, b=8, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -12 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √76) / 2.
5. Calculate roots: x1 = 0.358898943540674, x2 = -8.358898943540673.
Final Answe... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 9, 18, 3, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 9 + 18 + 3 + 14) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.4, -2.6, 6.4, -8.6, 2.4]
Squared Differences: [5.76, 6.76, 40.96, 73.96, 5.76]
... |
Compute the quotient and remainder of 10394 divided by 70. Show the long division steps. | To divide 10394 by 70:
1. Determine how many times 70 goes into the leading digits of 10394.
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: 10394 / 70 = 148 with a remainder of 34.
Proof: (148 * 70) ... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 8^2 = c^2
2. Calculate the squares: 324 + 64 = c^2
3. Sum the squares: 388 = c^2
4. Take the square root of both sides: c = √388
Final Ans... |
Add the fractions 5/2 and 12/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 5/2 by multiplying top and bottom by 7: 35/14.
3. Convert 12/14 by multiplying top and bottom by 1: 12/14.
4. Add the numerators: 35 + 12 = 47.
5. The resulting fraction is 47/14.
Final Answer: 4... |
Compute the quotient and remainder of 47497 divided by 52. Show the long division steps. | To divide 47497 by 52:
1. Determine how many times 52 goes into the leading digits of 47497.
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: 47497 / 52 = 913 with a remainder of 21.
Proof: (913 * 52) ... |
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 = 3 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: 3^2 + 7^2 = c^2
2. Calculate the squares: 9 + 49 = c^2
3. Sum the squares: 58 = c^2
4. Take the square root of both sides: c = √58
Final Answer: ... |
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(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 indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^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: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 9x^2 + 3x + -6 = 0 | 1. Identify coefficients: a=9, b=3, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -216 = 225.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √225) / 18.
5. Calculate roots: x1 = 0.6666666666666666, x2 = -1.0.
Final Answer: x = 0.6... |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[-4, -5]
B =
[3, -1]
[-1, 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) = (-2*3) + (0*-1) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (0*0) = 2
C21 = (A21 * B11) + (A22 * B21) = (-4*3) + (-5*-1) = -7
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (-5*0) = 4
Resul... |
Multiply the following 2x2 matrices:
A =
[4, -1]
[0, -4]
B =
[-3, 5]
[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) = (4*-3) + (-1*2) = -14
C12 = (A11 * B12) + (A12 * B22) = (4*5) + (-1*-4) = 24
C21 = (A21 * B11) + (A22 * B21) = (0*-3) + (-4*2) = -8
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (-4*-4) = 16
Resu... |
Compute the quotient and remainder of 43510 divided by 83. Show the long division steps. | To divide 43510 by 83:
1. Determine how many times 83 goes into the leading digits of 43510.
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: 43510 / 83 = 524 with a remainder of 18.
Proof: (524 * 83) ... |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 19^2 = c^2
2. Calculate the squares: 324 + 361 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
Final A... |
Find the roots of the quadratic equation: 7x^2 + -6x + -5 = 0 | 1. Identify coefficients: a=7, b=-6, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -140 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √176) / 14.
5. Calculate roots: x1 = 1.3761785115301142, x2 = -0.5190356543872571.
Final... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 12, 19, 3, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 12 + 19 + 3 + 7) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.2, 0.2, 7.2, -8.8, -4.8]
Squared Differences: [38.44, 0.04, 51.84, 77.44, 23.04]
... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 20^2 = c^2
2. Calculate the squares: 64 + 400 = c^2
3. Sum the squares: 464 = c^2
4. Take the square root of both sides: c = √464
Final Ans... |
Add the fractions 12/15 and 15/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 5, which is 15.
2. Convert 12/15 by multiplying top and bottom by 1: 12/15.
3. Convert 15/5 by multiplying top and bottom by 3: 45/15.
4. Add the numerators: 12 + 45 = 57.
5. The resulting fraction is 57/15.
6. Simplify by... |
Add the fractions 8/12 and 15/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 10, which is 60.
2. Convert 8/12 by multiplying top and bottom by 5: 40/60.
3. Convert 15/10 by multiplying top and bottom by 6: 90/60.
4. Add the numerators: 40 + 90 = 130.
5. The resulting fraction is 130/60.
6. Simplify... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[1, 0]
[-1, 4]
B =
[2, 0]
[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) = (1*2) + (0*5) = 2
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (0*2) = 0
C21 = (A21 * B11) + (A22 * B21) = (-1*2) + (4*5) = 18
C22 = (A21 * B12) + (A22 * B22) = (-1*0) + (4*2) = 8
Result Matrix ... |
Find the roots of the quadratic equation: 7x^2 + -10x + -6 = 0 | 1. Identify coefficients: a=7, b=-10, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -168 = 268.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √268) / 14.
5. Calculate roots: x1 = 1.8836218245532073, x2 = -0.4550503959817786.
Fi... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 8, 10, 1, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 8 + 10 + 1 + 19) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, -1.2, 0.8, -8.2, 9.8]
Squared Differences: [1.44, 1.44, 0.64, 67.24, 96.04]
... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 17^2 = c^2
2. Calculate the squares: 225 + 289 = c^2
3. Sum the squares: 514 = c^2
4. Take the square root of both sides: c = √514
Final A... |
Add the fractions 10/9 and 9/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 11, which is 99.
2. Convert 10/9 by multiplying top and bottom by 11: 110/99.
3. Convert 9/11 by multiplying top and bottom by 9: 81/99.
4. Add the numerators: 110 + 81 = 191.
5. The resulting fraction is 191/99.
Final Answ... |
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) = (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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 4, 11, 6, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 4 + 11 + 6 + 5) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.2, -4.8, 2.2, -2.8, -3.8]
Squared Differences: [84.64, 23.04, 4.84, 7.84, 14.44]
... |
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 sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + 7x + 5 = 0 | 1. Identify coefficients: a=8, b=7, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(8)(5) = 49 - 160 = -111.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √111i) / 16.
Final Answer: x = -0.4375 ± 0.6584783595532961i |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 19, 15, 8, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 19 + 15 + 8 + 12) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, 6.8, 2.8, -4.2, -0.2]
Squared Differences: [27.04, 46.24, 7.84, 17.64, 0.04]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 5, 16, 16, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 5 + 16 + 16 + 9) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, -5.0, 6.0, 6.0, -1.0]
Squared Differences: [36.0, 25.0, 36.0, 36.0, 1.0]
Su... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 3, 5, 18, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 3 + 5 + 18 + 19) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.4, -9.6, -7.6, 5.4, 6.4]
Squared Differences: [29.16, 92.16, 57.76, 29.16, 40.96]... |
Multiply the following 2x2 matrices:
A =
[3, -5]
[4, -4]
B =
[3, -1]
[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*3) + (-5*0) = 9
C12 = (A11 * B12) + (A12 * B22) = (3*-1) + (-5*2) = -13
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (-4*0) = 12
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (-4*2) = -12
Result... |
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 roots of the quadratic equation: 9x^2 + 4x + -4 = 0 | 1. Identify coefficients: a=9, b=4, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -144 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √160) / 18.
5. Calculate roots: x1 = 0.4805061467040843, x2 = -0.9249505911485288.
Final... |
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 +... |
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 +... |
Find the roots of the quadratic equation: 10x^2 + -4x + 9 = 0 | 1. Identify coefficients: a=10, b=-4, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(10)(9) = 16 - 360 = -344.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √344i) / 20.
Final Answer: x = 0.2 ± 0.9273618495495703i |
Compute the quotient and remainder of 24847 divided by 4. Show the long division steps. | To divide 24847 by 4:
1. Determine how many times 4 goes into the leading digits of 24847.
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: 24847 / 4 = 6211 with a remainder of 3.
Proof: (6211 * 4) + 3... |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^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: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[-1, 5]
B =
[3, 2]
[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*3) + (4*3) = 3
C12 = (A11 * B12) + (A12 * B22) = (-3*2) + (4*0) = -6
C21 = (A21 * B11) + (A22 * B21) = (-1*3) + (5*3) = 12
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (5*0) = -2
Result Mat... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 17, 18, 6, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 17 + 18 + 6 + 3) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.4, 5.4, 6.4, -5.6, -8.6]
Squared Differences: [5.76, 29.16, 40.96, 31.36, 73.96]
... |
Compute the quotient and remainder of 14034 divided by 33. Show the long division steps. | To divide 14034 by 33:
1. Determine how many times 33 goes into the leading digits of 14034.
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: 14034 / 33 = 425 with a remainder of 9.
Proof: (425 * 33) +... |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 20, 12, 15, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 20 + 12 + 15 + 19) / 5 = 81 / 5 = 16.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, 3.8, -4.2, -1.2, 2.8]
Squared Differences: [1.44, 14.44, 17.64, 1.44, 7.84]... |
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