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In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 14^2 = c^2
2. Calculate the squares: 49 + 196 = c^2
3. Sum the squares: 245 = c^2
4. Take the square root of both sides: c = √245
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 17, 10, 20, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 17 + 10 + 20 + 19) / 5 = 86 / 5 = 17.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -0.2, -7.2, 2.8, 1.8]
Squared Differences: [7.84, 0.04, 51.84, 7.84, 3.24]
... |
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 |
Multiply the following 2x2 matrices:
A =
[3, 1]
[4, 3]
B =
[2, 4]
[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) = (3*2) + (1*2) = 8
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (1*4) = 16
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (3*2) = 14
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (3*4) = 28
Result Matrix ... |
Compute the indefinite integral of f(x) = 4x^2. | To find the indefinite integral ∫ 4x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 4 / 3 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^2 dx = 4/3x^3 + C |
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) |
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) |
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 ... |
Compute the quotient and remainder of 47083 divided by 3. Show the long division steps. | To divide 47083 by 3:
1. Determine how many times 3 goes into the leading digits of 47083.
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: 47083 / 3 = 15694 with a remainder of 1.
Proof: (15694 * 3) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 12, 13, 11, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 12 + 13 + 11 + 7) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.2, 0.2, 1.2, -0.8, -4.8]
Squared Differences: [17.64, 0.04, 1.44, 0.64, 23.04]
... |
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 +... |
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(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 |
Add the fractions 13/5 and 13/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 7, which is 35.
2. Convert 13/5 by multiplying top and bottom by 7: 91/35.
3. Convert 13/7 by multiplying top and bottom by 5: 65/35.
4. Add the numerators: 91 + 65 = 156.
5. The resulting fraction is 156/35.
Final Answer: ... |
Find the roots of the quadratic equation: 8x^2 + -1x + 8 = 0 | 1. Identify coefficients: a=8, b=-1, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(8)(8) = 1 - 256 = -255.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √255i) / 16.
Final Answer: x = 0.0625 ± 0.998044963916957i |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 7^2 = c^2
2. Calculate the squares: 100 + 49 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
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) |
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 |
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 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 ... |
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) |
Add the fractions 14/10 and 1/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 8, which is 40.
2. Convert 14/10 by multiplying top and bottom by 4: 56/40.
3. Convert 1/8 by multiplying top and bottom by 5: 5/40.
4. Add the numerators: 56 + 5 = 61.
5. The resulting fraction is 61/40.
Final Answer: 61/... |
Compute the quotient and remainder of 23759 divided by 3. Show the long division steps. | To divide 23759 by 3:
1. Determine how many times 3 goes into the leading digits of 23759.
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: 23759 / 3 = 7919 with a remainder of 2.
Proof: (7919 * 3) + 2... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[0, 3]
B =
[-3, 4]
[5, -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*-3) + (2*5) = -5
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (2*-1) = 18
C21 = (A21 * B11) + (A22 * B21) = (0*-3) + (3*5) = 15
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (3*-1) = -3
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[-5, 2]
B =
[4, 0]
[-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) = (-3*4) + (5*-4) = -32
C12 = (A11 * B12) + (A12 * B22) = (-3*0) + (5*3) = 15
C21 = (A21 * B11) + (A22 * B21) = (-5*4) + (2*-4) = -28
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (2*3) = 6
Result... |
Compute the quotient and remainder of 75791 divided by 61. Show the long division steps. | To divide 75791 by 61:
1. Determine how many times 61 goes into the leading digits of 75791.
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: 75791 / 61 = 1242 with a remainder of 29.
Proof: (1242 * 61... |
Add the fractions 2/11 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 2/11 by multiplying top and bottom by 8: 16/88.
3. Convert 10/8 by multiplying top and bottom by 11: 110/88.
4. Add the numerators: 16 + 110 = 126.
5. The resulting fraction is 126/88.
6. Simplif... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 20, 1, 19, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 20 + 1 + 19 + 17) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.0, 8.0, -11.0, 7.0, 5.0]
Squared Differences: [81.0, 64.0, 121.0, 49.0, 25.0]
... |
Compute the quotient and remainder of 97122 divided by 95. Show the long division steps. | To divide 97122 by 95:
1. Determine how many times 95 goes into the leading digits of 97122.
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: 97122 / 95 = 1022 with a remainder of 32.
Proof: (1022 * 95... |
Find the roots of the quadratic equation: 6x^2 + 7x + 10 = 0 | 1. Identify coefficients: a=6, b=7, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(6)(10) = 49 - 240 = -191.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √191i) / 12.
Final Answer: x = -0.5833333333333334 ± 1.1516895800904379i |
Multiply the following 2x2 matrices:
A =
[5, -2]
[-5, 5]
B =
[-4, -1]
[5, 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*-4) + (-2*5) = -30
C12 = (A11 * B12) + (A12 * B22) = (5*-1) + (-2*1) = -7
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (5*5) = 45
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (5*1) = 10
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 11, 4, 4, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 11 + 4 + 4 + 16) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.2, 3.8, -3.2, -3.2, 8.8]
Squared Differences: [38.44, 14.44, 10.24, 10.24, 77.44]
... |
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 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: [8, 13, 5, 13, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 13 + 5 + 13 + 20) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, 1.2, -6.8, 1.2, 8.2]
Squared Differences: [14.44, 1.44, 46.24, 1.44, 67.24]
... |
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 = 12 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: 12^2 + 10^2 = c^2
2. Calculate the squares: 144 + 100 = c^2
3. Sum the squares: 244 = c^2
4. Take the square root of both sides: c = √244
Final A... |
Compute the quotient and remainder of 51706 divided by 28. Show the long division steps. | To divide 51706 by 28:
1. Determine how many times 28 goes into the leading digits of 51706.
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: 51706 / 28 = 1846 with a remainder of 18.
Proof: (1846 * 28... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 13, 5, 13, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 13 + 5 + 13 + 6) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.8, 1.8, -6.2, 1.8, -5.2]
Squared Differences: [60.84, 3.24, 38.44, 3.24, 27.04]
... |
Find the roots of the quadratic equation: 6x^2 + 3x + 8 = 0 | 1. Identify coefficients: a=6, b=3, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(6)(8) = 9 - 192 = -183.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √183i) / 12.
Final Answer: x = -0.25 ± 1.1273124382057236i |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Compute the 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 |
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 ... |
Compute the quotient and remainder of 18200 divided by 65. Show the long division steps. | To divide 18200 by 65:
1. Determine how many times 65 goes into the leading digits of 18200.
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: 18200 / 65 = 280 with a remainder of 0.
Proof: (280 * 65) +... |
Compute the quotient and remainder of 26596 divided by 74. Show the long division steps. | To divide 26596 by 74:
1. Determine how many times 74 goes into the leading digits of 26596.
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: 26596 / 74 = 359 with a remainder of 30.
Proof: (359 * 74) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 14, 11, 18, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 14 + 11 + 18 + 8) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.2, 2.8, -0.2, 6.8, -3.2]
Squared Differences: [38.44, 7.84, 0.04, 46.24, 10.24]
... |
Compute the quotient and remainder of 28506 divided by 24. Show the long division steps. | To divide 28506 by 24:
1. Determine how many times 24 goes into the leading digits of 28506.
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: 28506 / 24 = 1187 with a remainder of 18.
Proof: (1187 * 24... |
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 |
Add the fractions 5/2 and 9/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 8, which is 8.
2. Convert 5/2 by multiplying top and bottom by 4: 20/8.
3. Convert 9/8 by multiplying top and bottom by 1: 9/8.
4. Add the numerators: 20 + 9 = 29.
5. The resulting fraction is 29/8.
Final Answer: 29/8 |
Find the roots of the quadratic equation: 1x^2 + -5x + 5 = 0 | 1. Identify coefficients: a=1, b=-5, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 20 = 5.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √5) / 2.
5. Calculate roots: x1 = 3.618033988749895, x2 = 1.381966011250105.
Final Answer: x ... |
Compute the quotient and remainder of 71805 divided by 63. Show the long division steps. | To divide 71805 by 63:
1. Determine how many times 63 goes into the leading digits of 71805.
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: 71805 / 63 = 1139 with a remainder of 48.
Proof: (1139 * 63... |
Multiply the following 2x2 matrices:
A =
[3, 0]
[-5, 2]
B =
[5, 5]
[-2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*5) + (0*-2) = 15
C12 = (A11 * B12) + (A12 * B22) = (3*5) + (0*0) = 15
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (2*-2) = -29
C22 = (A21 * B12) + (A22 * B22) = (-5*5) + (2*0) = -25
Result ... |
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 |
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 + 8x + 1 = 0 | 1. Identify coefficients: a=2, b=8, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - 8 = 56.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √56) / 4.
5. Calculate roots: x1 = -0.12917130661302934, x2 = -3.8708286933869704.
Final Answ... |
Find the roots of the quadratic equation: 3x^2 + 10x + -7 = 0 | 1. Identify coefficients: a=3, b=10, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -84 = 184.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √184) / 6.
5. Calculate roots: x1 = 0.5941099943750894, x2 = -3.927443327708423.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 16^2 = c^2
2. Calculate the squares: 9 + 256 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Find the roots of the quadratic equation: 6x^2 + -1x + 1 = 0 | 1. Identify coefficients: a=6, b=-1, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(6)(1) = 1 - 24 = -23.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √23i) / 12.
Final Answer: x = 0.08333333333333333 ± 0.3996526269427266i |
Compute the quotient and remainder of 44520 divided by 31. Show the long division steps. | To divide 44520 by 31:
1. Determine how many times 31 goes into the leading digits of 44520.
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: 44520 / 31 = 1436 with a remainder of 4.
Proof: (1436 * 31)... |
Find the roots of the quadratic equation: 6x^2 + 10x + 9 = 0 | 1. Identify coefficients: a=6, b=10, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 10^2 - 4(6)(9) = 100 - 216 = -116.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-10 ± √116i) / 12.
Final Answer: x = -0.8333333333333334 ± 0.8975274678557507i |
Compute the quotient and remainder of 69522 divided by 49. Show the long division steps. | To divide 69522 by 49:
1. Determine how many times 49 goes into the leading digits of 69522.
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: 69522 / 49 = 1418 with a remainder of 40.
Proof: (1418 * 49... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 14^2 = c^2
2. Calculate the squares: 121 + 196 = c^2
3. Sum the squares: 317 = c^2
4. Take the square root of both sides: c = √317
Final A... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[-3, -4]
B =
[-5, 4]
[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*-5) + (0*2) = -20
C12 = (A11 * B12) + (A12 * B22) = (4*4) + (0*-4) = 16
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-4*2) = 7
C22 = (A21 * B12) + (A22 * B22) = (-3*4) + (-4*-4) = 4
Result... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 15^2 = c^2
2. Calculate the squares: 81 + 225 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
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 |
Add the fractions 4/8 and 1/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 12, which is 24.
2. Convert 4/8 by multiplying top and bottom by 3: 12/24.
3. Convert 1/12 by multiplying top and bottom by 2: 2/24.
4. Add the numerators: 12 + 2 = 14.
5. The resulting fraction is 14/24.
6. Simplify by div... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 6, 18, 14, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 6 + 18 + 14 + 20) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, -7.8, 4.2, 0.2, 6.2]
Squared Differences: [7.84, 60.84, 17.64, 0.04, 38.44]
... |
Compute the quotient and remainder of 45348 divided by 21. Show the long division steps. | To divide 45348 by 21:
1. Determine how many times 21 goes into the leading digits of 45348.
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: 45348 / 21 = 2159 with a remainder of 9.
Proof: (2159 * 21)... |
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) |
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 28057 divided by 61. Show the long division steps. | To divide 28057 by 61:
1. Determine how many times 61 goes into the leading digits of 28057.
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: 28057 / 61 = 459 with a remainder of 58.
Proof: (459 * 61) ... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[4, 1]
B =
[-2, -1]
[3, -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) = (4*-2) + (0*3) = -8
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (0*-1) = -4
C21 = (A21 * B11) + (A22 * B21) = (4*-2) + (1*3) = -5
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (1*-1) = -5
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 6^2 = c^2
2. Calculate the squares: 400 + 36 = c^2
3. Sum the squares: 436 = c^2
4. Take the square root of both sides: c = √436
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) |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-2, 0]
B =
[5, -2]
[1, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*5) + (3*1) = 13
C12 = (A11 * B12) + (A12 * B22) = (2*-2) + (3*5) = 11
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (0*1) = -10
C22 = (A21 * B12) + (A22 * B22) = (-2*-2) + (0*5) = 4
Result Ma... |
Add the fractions 14/5 and 15/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 2, which is 10.
2. Convert 14/5 by multiplying top and bottom by 2: 28/10.
3. Convert 15/2 by multiplying top and bottom by 5: 75/10.
4. Add the numerators: 28 + 75 = 103.
5. The resulting fraction is 103/10.
Final Answer: ... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 20, 18, 15, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 20 + 18 + 15 + 11) / 5 = 70 / 5 = 14.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.0, 6.0, 4.0, 1.0, -3.0]
Squared Differences: [64.0, 36.0, 16.0, 1.0, 9.0]
Su... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 9^2 = c^2
2. Calculate the squares: 361 + 81 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 12^2 = c^2
2. Calculate the squares: 36 + 144 = c^2
3. Sum the squares: 180 = c^2
4. Take the square root of both sides: c = √180
Final Ans... |
Compute the quotient and remainder of 49617 divided by 7. Show the long division steps. | To divide 49617 by 7:
1. Determine how many times 7 goes into the leading digits of 49617.
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: 49617 / 7 = 7088 with a remainder of 1.
Proof: (7088 * 7) + 1... |
Add the fractions 11/6 and 1/2. 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 11/6 by multiplying top and bottom by 1: 11/6.
3. Convert 1/2 by multiplying top and bottom by 3: 3/6.
4. Add the numerators: 11 + 3 = 14.
5. The resulting fraction is 14/6.
6. Simplify by dividing... |
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 |
Add the fractions 5/8 and 2/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 14, which is 56.
2. Convert 5/8 by multiplying top and bottom by 7: 35/56.
3. Convert 2/14 by multiplying top and bottom by 4: 8/56.
4. Add the numerators: 35 + 8 = 43.
5. The resulting fraction is 43/56.
Final Answer: 43/5... |
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: [3, 3, 7, 16, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 3 + 7 + 16 + 2) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -3.2, 0.8, 9.8, -4.2]
Squared Differences: [10.24, 10.24, 0.64, 96.04, 17.64]
... |
Find the roots of the quadratic equation: 8x^2 + -5x + -5 = 0 | 1. Identify coefficients: a=8, b=-5, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -160 = 185.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √185) / 16.
5. Calculate roots: x1 = 1.1625919067959654, x2 = -0.5375919067959652.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 11^2 = c^2
2. Calculate the squares: 100 + 121 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 18, 4, 9, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 18 + 4 + 9 + 11) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, 9.4, -4.6, 0.4, 2.4]
Squared Differences: [57.76, 88.36, 21.16, 0.16, 5.76]
... |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Add the fractions 12/15 and 7/15. 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 12/15 by multiplying top and bottom by 1: 12/15.
3. Convert 7/15 by multiplying top and bottom by 1: 7/15.
4. Add the numerators: 12 + 7 = 19.
5. The resulting fraction is 19/15.
Final Answer: 1... |
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 ... |
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 |
Add the fractions 7/6 and 2/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 7, which is 42.
2. Convert 7/6 by multiplying top and bottom by 7: 49/42.
3. Convert 2/7 by multiplying top and bottom by 6: 12/42.
4. Add the numerators: 49 + 12 = 61.
5. The resulting fraction is 61/42.
Final Answer: 61/4... |
Add the fractions 3/7 and 4/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 12, which is 84.
2. Convert 3/7 by multiplying top and bottom by 12: 36/84.
3. Convert 4/12 by multiplying top and bottom by 7: 28/84.
4. Add the numerators: 36 + 28 = 64.
5. The resulting fraction is 64/84.
6. Simplify by ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 17^2 = c^2
2. Calculate the squares: 256 + 289 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
Compute the quotient and remainder of 5252 divided by 81. Show the long division steps. | To divide 5252 by 81:
1. Determine how many times 81 goes into the leading digits of 5252.
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: 5252 / 81 = 64 with a remainder of 68.
Proof: (64 * 81) + 68 ... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 10x^2 + 0x + -5 = 0 | 1. Identify coefficients: a=10, b=0, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -200 = 200.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √200) / 20.
5. Calculate roots: x1 = 0.7071067811865476, x2 = -0.7071067811865476.
Final ... |
Find the roots of the quadratic equation: 6x^2 + 4x + 8 = 0 | 1. Identify coefficients: a=6, b=4, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(6)(8) = 16 - 192 = -176.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √176i) / 12.
Final Answer: x = -0.3333333333333333 ± 1.1055415967851332i |
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