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In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 19^2 = c^2
2. Calculate the squares: 121 + 361 = c^2
3. Sum the squares: 482 = c^2
4. Take the square root of both sides: c = √482
Final A... |
Compute the quotient and remainder of 22474 divided by 56. Show the long division steps. | To divide 22474 by 56:
1. Determine how many times 56 goes into the leading digits of 22474.
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: 22474 / 56 = 401 with a remainder of 18.
Proof: (401 * 56) ... |
Add the fractions 6/11 and 15/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 14, which is 154.
2. Convert 6/11 by multiplying top and bottom by 14: 84/154.
3. Convert 15/14 by multiplying top and bottom by 11: 165/154.
4. Add the numerators: 84 + 165 = 249.
5. The resulting fraction is 249/154.
Fin... |
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 11/9 and 8/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 15, which is 45.
2. Convert 11/9 by multiplying top and bottom by 5: 55/45.
3. Convert 8/15 by multiplying top and bottom by 3: 24/45.
4. Add the numerators: 55 + 24 = 79.
5. The resulting fraction is 79/45.
Final Answer: 7... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Compute the indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 13, 5, 12, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 13 + 5 + 12 + 9) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, 3.6, -4.4, 2.6, -0.4]
Squared Differences: [1.96, 12.96, 19.36, 6.76, 0.16]
... |
Multiply the following 2x2 matrices:
A =
[-1, -2]
[1, 1]
B =
[-1, -5]
[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) = (-1*-1) + (-2*5) = -9
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (-2*5) = -5
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (1*5) = 4
C22 = (A21 * B12) + (A22 * B22) = (1*-5) + (1*5) = 0
Result ... |
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: [11, 8, 16, 16, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 8 + 16 + 16 + 5) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, -3.2, 4.8, 4.8, -6.2]
Squared Differences: [0.04, 10.24, 23.04, 23.04, 38.44]... |
Compute the quotient and remainder of 13177 divided by 30. Show the long division steps. | To divide 13177 by 30:
1. Determine how many times 30 goes into the leading digits of 13177.
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: 13177 / 30 = 439 with a remainder of 7.
Proof: (439 * 30) +... |
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 |
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 |
Compute the quotient and remainder of 33030 divided by 27. Show the long division steps. | To divide 33030 by 27:
1. Determine how many times 27 goes into the leading digits of 33030.
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: 33030 / 27 = 1223 with a remainder of 9.
Proof: (1223 * 27)... |
Add the fractions 2/13 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 2/13 by multiplying top and bottom by 1: 2/13.
3. Convert 6/13 by multiplying top and bottom by 1: 6/13.
4. Add the numerators: 2 + 6 = 8.
5. The resulting fraction is 8/13.
Final Answer: 8/13 |
Compute the quotient and remainder of 64232 divided by 76. Show the long division steps. | To divide 64232 by 76:
1. Determine how many times 76 goes into the leading digits of 64232.
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: 64232 / 76 = 845 with a remainder of 12.
Proof: (845 * 76) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 5, 7, 16, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 5 + 7 + 16 + 13) / 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, -6.8, -4.8, 4.2, 1.2]
Squared Differences: [38.44, 46.24, 23.04, 17.64, 1.44]
... |
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 |
Add the fractions 2/7 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 13, which is 91.
2. Convert 2/7 by multiplying top and bottom by 13: 26/91.
3. Convert 5/13 by multiplying top and bottom by 7: 35/91.
4. Add the numerators: 26 + 35 = 61.
5. The resulting fraction is 61/91.
Final Answer: 6... |
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: [5, 17, 6, 14, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 17 + 6 + 14 + 6) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, 7.4, -3.6, 4.4, -3.6]
Squared Differences: [21.16, 54.76, 12.96, 19.36, 12.96]
... |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^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: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Find the roots of the quadratic equation: 2x^2 + -8x + -6 = 0 | 1. Identify coefficients: a=2, b=-8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -48 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √112) / 4.
5. Calculate roots: x1 = 4.645751311064591, x2 = -0.6457513110645907.
Final An... |
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 ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 9^2 = c^2
2. Calculate the squares: 25 + 81 = c^2
3. Sum the squares: 106 = c^2
4. Take the square root of both sides: c = √106
Final Answe... |
Add the fractions 11/15 and 9/15. 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 11/15 by multiplying top and bottom by 1: 11/15.
3. Convert 9/15 by multiplying top and bottom by 1: 9/15.
4. Add the numerators: 11 + 9 = 20.
5. The resulting fraction is 20/15.
6. Simplify by ... |
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) |
Find the roots of the quadratic equation: 2x^2 + 9x + -9 = 0 | 1. Identify coefficients: a=2, b=9, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -72 = 153.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √153) / 4.
5. Calculate roots: x1 = 0.8423292192132452, x2 = -5.342329219213245.
Final An... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 4^2 = c^2
2. Calculate the squares: 324 + 16 = c^2
3. Sum the squares: 340 = c^2
4. Take the square root of both sides: c = √340
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 16^2 = c^2
2. Calculate the squares: 289 + 256 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
Multiply the following 2x2 matrices:
A =
[-5, 2]
[3, 5]
B =
[-2, 3]
[0, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-2) + (2*0) = 10
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (2*-1) = -17
C21 = (A21 * B11) + (A22 * B21) = (3*-2) + (5*0) = -6
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (5*-1) = 4
Result ... |
Find the roots of the quadratic equation: 9x^2 + 1x + 8 = 0 | 1. Identify coefficients: a=9, b=1, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(9)(8) = 1 - 288 = -287.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √287i) / 18.
Final Answer: x = -0.05555555555555555 ± 0.941170797005412i |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Multiply the following 2x2 matrices:
A =
[4, 1]
[-2, -1]
B =
[0, -4]
[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) = (4*0) + (1*1) = 1
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (1*0) = -16
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-1*1) = -1
C22 = (A21 * B12) + (A22 * B22) = (-2*-4) + (-1*0) = 8
Result M... |
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 ... |
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) |
Multiply the following 2x2 matrices:
A =
[-5, 1]
[5, -5]
B =
[-2, 2]
[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) = (-5*-2) + (1*4) = 14
C12 = (A11 * B12) + (A12 * B22) = (-5*2) + (1*3) = -7
C21 = (A21 * B11) + (A22 * B21) = (5*-2) + (-5*4) = -30
C22 = (A21 * B12) + (A22 * B22) = (5*2) + (-5*3) = -5
Result... |
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 |
Compute the quotient and remainder of 77831 divided by 55. Show the long division steps. | To divide 77831 by 55:
1. Determine how many times 55 goes into the leading digits of 77831.
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: 77831 / 55 = 1415 with a remainder of 6.
Proof: (1415 * 55)... |
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 + 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 ... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 9^2 = c^2
2. Calculate the squares: 64 + 81 = c^2
3. Sum the squares: 145 = c^2
4. Take the square root of both sides: c = √145
Final Answe... |
Find the roots of the quadratic equation: 5x^2 + -2x + -3 = 0 | 1. Identify coefficients: a=5, b=-2, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -60 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √64) / 10.
5. Calculate roots: x1 = 1.0, x2 = -0.6.
Final Answer: x = 1.0, -0.6 |
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... |
Find the roots of the quadratic equation: 4x^2 + 4x + -3 = 0 | 1. Identify coefficients: a=4, b=4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -48 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √64) / 8.
5. Calculate roots: x1 = 0.5, x2 = -1.5.
Final Answer: x = 0.5, -1.5 |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 9x^2 + 6x + -9 = 0 | 1. Identify coefficients: a=9, b=6, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -324 = 360.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √360) / 18.
5. Calculate roots: x1 = 0.7207592200561265, x2 = -1.387425886722793.
Final ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 6^2 = c^2
2. Calculate the squares: 256 + 36 = c^2
3. Sum the squares: 292 = c^2
4. Take the square root of both sides: c = √292
Final Ans... |
Compute the indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 15^2 = c^2
2. Calculate the squares: 64 + 225 = c^2
3. Sum the squares: 289 = c^2
4. Take the square root of both sides: c = √289
Final Ans... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (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 +... |
Multiply the following 2x2 matrices:
A =
[3, -5]
[-3, 4]
B =
[-2, -2]
[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) = (3*-2) + (-5*5) = -31
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (-5*-4) = 14
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (4*5) = 26
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (4*-4) = -10
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 17, 13, 8, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 17 + 13 + 8 + 2) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.6, 5.6, 1.6, -3.4, -9.4]
Squared Differences: [31.36, 31.36, 2.56, 11.56, 88.36]
... |
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) |
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 |
Find the roots of the quadratic equation: 9x^2 + 4x + -6 = 0 | 1. Identify coefficients: a=9, b=4, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -216 = 232.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √232) / 18.
5. Calculate roots: x1 = 0.6239747895404343, x2 = -1.0684192339848788.
Final... |
Find the roots of the quadratic equation: 5x^2 + -7x + 3 = 0 | 1. Identify coefficients: a=5, b=-7, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(5)(3) = 49 - 60 = -11.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √11i) / 10.
Final Answer: x = 0.7 ± 0.33166247903553997i |
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 roots of the quadratic equation: 7x^2 + -2x + -3 = 0 | 1. Identify coefficients: a=7, b=-2, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -84 = 88.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √88) / 14.
5. Calculate roots: x1 = 0.8129165371176328, x2 = -0.5272022514033471.
Final Ans... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 12, 15, 5, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 12 + 15 + 5 + 10) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.2, 0.2, 3.2, -6.8, -1.8]
Squared Differences: [27.04, 0.04, 10.24, 46.24, 3.24]
... |
Find the roots of the quadratic equation: 10x^2 + -7x + 2 = 0 | 1. Identify coefficients: a=10, b=-7, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(10)(2) = 49 - 80 = -31.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √31i) / 20.
Final Answer: x = 0.35 ± 0.27838821814150105i |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 3, 12, 14, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 3 + 12 + 14 + 11) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, -7.2, 1.8, 3.8, 0.8]
Squared Differences: [0.64, 51.84, 3.24, 14.44, 0.64]
... |
Find the roots of the quadratic equation: 7x^2 + -2x + -9 = 0 | 1. Identify coefficients: a=7, b=-2, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -252 = 256.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √256) / 14.
5. Calculate roots: x1 = 1.2857142857142858, x2 = -1.0.
Final Answer: x = 1.2... |
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 tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 5x^2 + -2x + 0 = 0 | 1. Identify coefficients: a=5, b=-2, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - 0 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √4) / 10.
5. Calculate roots: x1 = 0.4, x2 = 0.0.
Final Answer: x = 0.4, 0.0 |
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) |
Find the roots of the quadratic equation: 2x^2 + -3x + -9 = 0 | 1. Identify coefficients: a=2, b=-3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -72 = 81.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √81) / 4.
5. Calculate roots: x1 = 3.0, x2 = -1.5.
Final Answer: x = 3.0, -1.5 |
Compute the quotient and remainder of 80154 divided by 32. Show the long division steps. | To divide 80154 by 32:
1. Determine how many times 32 goes into the leading digits of 80154.
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: 80154 / 32 = 2504 with a remainder of 26.
Proof: (2504 * 32... |
Find the roots of the quadratic equation: 4x^2 + -5x + 1 = 0 | 1. Identify coefficients: a=4, b=-5, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 16 = 9.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √9) / 8.
5. Calculate roots: x1 = 1.0, x2 = 0.25.
Final Answer: x = 1.0, 0.25 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 4^2 = c^2
2. Calculate the squares: 144 + 16 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
Final Ans... |
Compute the quotient and remainder of 31185 divided by 87. Show the long division steps. | To divide 31185 by 87:
1. Determine how many times 87 goes into the leading digits of 31185.
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: 31185 / 87 = 358 with a remainder of 39.
Proof: (358 * 87) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 13, 8, 5, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 13 + 8 + 5 + 6) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.8, 2.8, -2.2, -5.2, -4.2]
Squared Differences: [77.44, 7.84, 4.84, 27.04, 17.64]
... |
Find the roots of the quadratic equation: 3x^2 + 7x + -4 = 0 | 1. Identify coefficients: a=3, b=7, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -48 = 97.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √97) / 6.
5. Calculate roots: x1 = 0.47480963363268397, x2 = -2.808142966966017.
Final Ans... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^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: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
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 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 20^2 = c^2
2. Calculate the squares: 121 + 400 = c^2
3. Sum the squares: 521 = c^2
4. Take the square root of both sides: c = √521
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 11^2 = c^2
2. Calculate the squares: 400 + 121 = c^2
3. Sum the squares: 521 = c^2
4. Take the square root of both sides: c = √521
Final A... |
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 |
Compute the indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 5, 11, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 5 + 11 + 9 + 10) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, -4.4, 1.6, -0.4, 0.6]
Squared Differences: [6.76, 19.36, 2.56, 0.16, 0.36]
S... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 12^2 = c^2
2. Calculate the squares: 64 + 144 = c^2
3. Sum the squares: 208 = c^2
4. Take the square root of both sides: c = √208
Final Ans... |
Add the fractions 4/7 and 1/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 7, which is 7.
2. Convert 4/7 by multiplying top and bottom by 1: 4/7.
3. Convert 1/7 by multiplying top and bottom by 1: 1/7.
4. Add the numerators: 4 + 1 = 5.
5. The resulting fraction is 5/7.
Final Answer: 5/7 |
Add the fractions 9/15 and 7/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 8, which is 120.
2. Convert 9/15 by multiplying top and bottom by 8: 72/120.
3. Convert 7/8 by multiplying top and bottom by 15: 105/120.
4. Add the numerators: 72 + 105 = 177.
5. The resulting fraction is 177/120.
6. Simp... |
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 |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^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: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 3^2 = c^2
2. Calculate the squares: 256 + 9 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Compute the quotient and remainder of 36249 divided by 58. Show the long division steps. | To divide 36249 by 58:
1. Determine how many times 58 goes into the leading digits of 36249.
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: 36249 / 58 = 624 with a remainder of 57.
Proof: (624 * 58) ... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 54582 divided by 39. Show the long division steps. | To divide 54582 by 39:
1. Determine how many times 39 goes into the leading digits of 54582.
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: 54582 / 39 = 1399 with a remainder of 21.
Proof: (1399 * 39... |
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