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In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 6^2 = c^2
2. Calculate the squares: 100 + 36 = c^2
3. Sum the squares: 136 = c^2
4. Take the square root of both sides: c = √136
Final Ans... |
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 |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[2, -1]
B =
[-1, -1]
[2, -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) = (-4*-1) + (-5*2) = -6
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (-5*-2) = 14
C21 = (A21 * B11) + (A22 * B21) = (2*-1) + (-1*2) = -4
C22 = (A21 * B12) + (A22 * B22) = (2*-1) + (-1*-2) = 0
Re... |
Find the roots of the quadratic equation: 6x^2 + 7x + 1 = 0 | 1. Identify coefficients: a=6, b=7, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 24 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √25) / 12.
5. Calculate roots: x1 = -0.16666666666666666, x2 = -1.0.
Final Answer: x = -0.16... |
Compute the quotient and remainder of 75949 divided by 13. Show the long division steps. | To divide 75949 by 13:
1. Determine how many times 13 goes into the leading digits of 75949.
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: 75949 / 13 = 5842 with a remainder of 3.
Proof: (5842 * 13)... |
Find the roots of the quadratic equation: 5x^2 + -5x + 3 = 0 | 1. Identify coefficients: a=5, b=-5, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(5)(3) = 25 - 60 = -35.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √35i) / 10.
Final Answer: x = 0.5 ± 0.5916079783099616i |
Multiply the following 2x2 matrices:
A =
[2, 2]
[5, 0]
B =
[1, -4]
[-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*1) + (2*-1) = 0
C12 = (A11 * B12) + (A12 * B22) = (2*-4) + (2*-5) = -18
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (0*-1) = 5
C22 = (A21 * B12) + (A22 * B22) = (5*-4) + (0*-5) = -20
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 11^2 = c^2
2. Calculate the squares: 81 + 121 = c^2
3. Sum the squares: 202 = c^2
4. Take the square root of both sides: c = √202
Final Ans... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, -5]
[4, 2]
B =
[4, -3]
[-2, -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) = (-3*4) + (-5*-2) = -2
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (-5*-5) = 34
C21 = (A21 * B11) + (A22 * B21) = (4*4) + (2*-2) = 12
C22 = (A21 * B12) + (A22 * B22) = (4*-3) + (2*-5) = -22
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 5, 17, 5, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 5 + 17 + 5 + 8) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, -3.8, 8.2, -3.8, -0.8]
Squared Differences: [0.04, 14.44, 67.24, 14.44, 0.64]
... |
Add the fractions 12/5 and 10/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 11, which is 55.
2. Convert 12/5 by multiplying top and bottom by 11: 132/55.
3. Convert 10/11 by multiplying top and bottom by 5: 50/55.
4. Add the numerators: 132 + 50 = 182.
5. The resulting fraction is 182/55.
Final Ans... |
Add the fractions 6/8 and 10/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 10, which is 40.
2. Convert 6/8 by multiplying top and bottom by 5: 30/40.
3. Convert 10/10 by multiplying top and bottom by 4: 40/40.
4. Add the numerators: 30 + 40 = 70.
5. The resulting fraction is 70/40.
6. Simplify by ... |
Compute the indefinite integral of f(x) = 10x^2. | To find the indefinite integral ∫ 10x^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: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
Find the roots of the quadratic equation: 8x^2 + -10x + 10 = 0 | 1. Identify coefficients: a=8, b=-10, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(8)(10) = 100 - 320 = -220.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √220i) / 16.
Final Answer: x = 0.625 ± 0.9270248108869579i |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (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) |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 14, 12, 14, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 14 + 12 + 14 + 4) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.6, 2.6, 0.6, 2.6, -7.4]
Squared Differences: [2.56, 6.76, 0.36, 6.76, 54.76]
... |
Add the fractions 1/15 and 6/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 7, which is 105.
2. Convert 1/15 by multiplying top and bottom by 7: 7/105.
3. Convert 6/7 by multiplying top and bottom by 15: 90/105.
4. Add the numerators: 7 + 90 = 97.
5. The resulting fraction is 97/105.
Final Answer:... |
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 |
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... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 7^2 = c^2
2. Calculate the squares: 289 + 49 = c^2
3. Sum the squares: 338 = c^2
4. Take the square root of both sides: c = √338
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 16, 15, 12, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 16 + 15 + 12 + 1) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 6.6, 5.6, 2.6, -8.4]
Squared Differences: [40.96, 43.56, 31.36, 6.76, 70.56]
... |
Add the fractions 2/3 and 3/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 2/3 by multiplying top and bottom by 5: 10/15.
3. Convert 3/5 by multiplying top and bottom by 3: 9/15.
4. Add the numerators: 10 + 9 = 19.
5. The resulting fraction is 19/15.
Final Answer: 19/15 |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Find the roots of the quadratic equation: 1x^2 + 10x + -2 = 0 | 1. Identify coefficients: a=1, b=10, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -8 = 108.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √108) / 2.
5. Calculate roots: x1 = 0.19615242270663202, x2 = -10.196152422706632.
Fina... |
Find the roots of the quadratic equation: 9x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=9, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -36 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √100) / 18.
5. Calculate roots: x1 = 1.0, x2 = -0.1111111111111111.
Final Answer: x = 1.0... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, 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 +... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 11^2 = c^2
2. Calculate the squares: 169 + 121 = c^2
3. Sum the squares: 290 = c^2
4. Take the square root of both sides: c = √290
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 12^2 = c^2
2. Calculate the squares: 256 + 144 = c^2
3. Sum the squares: 400 = c^2
4. Take the square root of both sides: c = √400
Final A... |
Compute the quotient and remainder of 69709 divided by 84. Show the long division steps. | To divide 69709 by 84:
1. Determine how many times 84 goes into the leading digits of 69709.
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: 69709 / 84 = 829 with a remainder of 73.
Proof: (829 * 84) ... |
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 |
Compute the quotient and remainder of 9125 divided by 61. Show the long division steps. | To divide 9125 by 61:
1. Determine how many times 61 goes into the leading digits of 9125.
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: 9125 / 61 = 149 with a remainder of 36.
Proof: (149 * 61) + 3... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
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 |
Multiply the following 2x2 matrices:
A =
[1, -3]
[0, 2]
B =
[-5, 5]
[-4, -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*-5) + (-3*-4) = 7
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (-3*-1) = 8
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (2*-4) = -8
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (2*-1) = -2
Result ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 9, 20, 11, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 9 + 20 + 11 + 12) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.0, -2.0, 9.0, 0.0, 1.0]
Squared Differences: [64.0, 4.0, 81.0, 0.0, 1.0]
Sum ... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, -1]
[4, 2]
B =
[-5, -1]
[-4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-5) + (-1*-4) = 9
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (-1*5) = -4
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (2*-4) = -28
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (2*5) = 6
Resu... |
Find the roots of the quadratic equation: 3x^2 + -6x + -10 = 0 | 1. Identify coefficients: a=3, b=-6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -120 = 156.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √156) / 6.
5. Calculate roots: x1 = 3.0816659994661326, x2 = -1.0816659994661328.
Final... |
Find the roots of the quadratic equation: 4x^2 + 10x + 5 = 0 | 1. Identify coefficients: a=4, b=10, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 80 = 20.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √20) / 8.
5. Calculate roots: x1 = -0.6909830056250525, x2 = -1.8090169943749475.
Final A... |
Multiply the following 2x2 matrices:
A =
[4, 1]
[-1, -3]
B =
[-2, 2]
[4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-2) + (1*4) = -4
C12 = (A11 * B12) + (A12 * B22) = (4*2) + (1*5) = 13
C21 = (A21 * B11) + (A22 * B21) = (-1*-2) + (-3*4) = -10
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (-3*5) = -17
Resul... |
Multiply the following 2x2 matrices:
A =
[-4, 4]
[0, 2]
B =
[-1, -5]
[5, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-1) + (4*5) = 24
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (4*-4) = 4
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (2*5) = 10
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (2*-4) = -8
Result... |
Compute the quotient and remainder of 92130 divided by 65. Show the long division steps. | To divide 92130 by 65:
1. Determine how many times 65 goes into the leading digits of 92130.
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: 92130 / 65 = 1417 with a remainder of 25.
Proof: (1417 * 65... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Compute the indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^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: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + 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 |
Compute the quotient and remainder of 53147 divided by 58. Show the long division steps. | To divide 53147 by 58:
1. Determine how many times 58 goes into the leading digits of 53147.
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: 53147 / 58 = 916 with a remainder of 19.
Proof: (916 * 58) ... |
Add the fractions 4/9 and 5/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 14, which is 126.
2. Convert 4/9 by multiplying top and bottom by 14: 56/126.
3. Convert 5/14 by multiplying top and bottom by 9: 45/126.
4. Add the numerators: 56 + 45 = 101.
5. The resulting fraction is 101/126.
Final Ans... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 3^2 = c^2
2. Calculate the squares: 324 + 9 = c^2
3. Sum the squares: 333 = c^2
4. Take the square root of both sides: c = √333
Final Answ... |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 20^2 = c^2
2. Calculate the squares: 144 + 400 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Multiply the following 2x2 matrices:
A =
[-4, 4]
[2, 2]
B =
[4, 3]
[-3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*4) + (4*-3) = -28
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (4*-4) = -28
C21 = (A21 * B11) + (A22 * B21) = (2*4) + (2*-3) = 2
C22 = (A21 * B12) + (A22 * B22) = (2*3) + (2*-4) = -2
Result... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 12^2 = c^2
2. Calculate the squares: 324 + 144 = c^2
3. Sum the squares: 468 = c^2
4. Take the square root of both sides: c = √468
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 15, 13, 19, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 15 + 13 + 19 + 7) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.2, 3.8, 1.8, 7.8, -4.2]
Squared Differences: [84.64, 14.44, 3.24, 60.84, 17.64]
... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 20^2 = c^2
2. Calculate the squares: 36 + 400 = c^2
3. Sum the squares: 436 = c^2
4. Take the square root of both sides: c = √436
Final Ans... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Find the 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 4, 20, 14, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 4 + 20 + 14 + 2) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.0, -8.0, 8.0, 2.0, -10.0]
Squared Differences: [64.0, 64.0, 64.0, 4.0, 100.0]
... |
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 |
Add the fractions 14/6 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 3, which is 6.
2. Convert 14/6 by multiplying top and bottom by 1: 14/6.
3. Convert 14/3 by multiplying top and bottom by 2: 28/6.
4. Add the numerators: 14 + 28 = 42.
5. The resulting fraction is 42/6.
6. Simplify by divid... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
Compute the quotient and remainder of 67311 divided by 32. Show the long division steps. | To divide 67311 by 32:
1. Determine how many times 32 goes into the leading digits of 67311.
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: 67311 / 32 = 2103 with a remainder of 15.
Proof: (2103 * 32... |
Add the fractions 8/11 and 7/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 12, which is 132.
2. Convert 8/11 by multiplying top and bottom by 12: 96/132.
3. Convert 7/12 by multiplying top and bottom by 11: 77/132.
4. Add the numerators: 96 + 77 = 173.
5. The resulting fraction is 173/132.
Final ... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 4, 2, 15, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 4 + 2 + 15 + 6) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, -4.2, -6.2, 6.8, -2.2]
Squared Differences: [33.64, 17.64, 38.44, 46.24, 4.84]
... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 8^2 = c^2
2. Calculate the squares: 100 + 64 = c^2
3. Sum the squares: 164 = c^2
4. Take the square root of both sides: c = √164
Final Ans... |
Add the fractions 11/2 and 13/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 10, which is 10.
2. Convert 11/2 by multiplying top and bottom by 5: 55/10.
3. Convert 13/10 by multiplying top and bottom by 1: 13/10.
4. Add the numerators: 55 + 13 = 68.
5. The resulting fraction is 68/10.
6. Simplify by... |
Multiply the following 2x2 matrices:
A =
[0, 2]
[-1, 4]
B =
[0, -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) = (0*0) + (2*3) = 6
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (2*-1) = -2
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (4*3) = 12
C22 = (A21 * B12) + (A22 * B22) = (-1*-1) + (4*-1) = -3
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 10^2 = c^2
2. Calculate the squares: 169 + 100 = c^2
3. Sum the squares: 269 = c^2
4. Take the square root of both sides: c = √269
Final A... |
Compute the quotient and remainder of 72218 divided by 75. Show the long division steps. | To divide 72218 by 75:
1. Determine how many times 75 goes into the leading digits of 72218.
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: 72218 / 75 = 962 with a remainder of 68.
Proof: (962 * 75) ... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 12^2 = c^2
2. Calculate the squares: 144 + 144 = c^2
3. Sum the squares: 288 = c^2
4. Take the square root of both sides: c = √288
Final A... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
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 |
Compute the quotient and remainder of 21959 divided by 73. Show the long division steps. | To divide 21959 by 73:
1. Determine how many times 73 goes into the leading digits of 21959.
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: 21959 / 73 = 300 with a remainder of 59.
Proof: (300 * 73) ... |
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: 5x^2 + -4x + -10 = 0 | 1. Identify coefficients: a=5, b=-4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -200 = 216.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √216) / 10.
5. Calculate roots: x1 = 1.8696938456699068, x2 = -1.069693845669907.
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 |
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 + 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 ... |
Multiply the following 2x2 matrices:
A =
[-5, -5]
[3, -5]
B =
[0, 4]
[-4, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*0) + (-5*-4) = 20
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (-5*-4) = 0
C21 = (A21 * B11) + (A22 * B21) = (3*0) + (-5*-4) = 20
C22 = (A21 * B12) + (A22 * B22) = (3*4) + (-5*-4) = 32
Resu... |
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) |
Compute the quotient and remainder of 88300 divided by 15. Show the long division steps. | To divide 88300 by 15:
1. Determine how many times 15 goes into the leading digits of 88300.
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: 88300 / 15 = 5886 with a remainder of 10.
Proof: (5886 * 15... |
Find the roots of the quadratic equation: 8x^2 + -6x + 1 = 0 | 1. Identify coefficients: a=8, b=-6, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 32 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √4) / 16.
5. Calculate roots: x1 = 0.5, x2 = 0.25.
Final Answer: x = 0.5, 0.25 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 20^2 = c^2
2. Calculate the squares: 400 + 400 = c^2
3. Sum the squares: 800 = c^2
4. Take the square root of both sides: c = √800
Final A... |
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 ... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[5, 0]
B =
[2, -3]
[3, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*2) + (-1*3) = -11
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-1*2) = 10
C21 = (A21 * B11) + (A22 * B21) = (5*2) + (0*3) = 10
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (0*2) = -15
Resul... |
Compute the quotient and remainder of 54712 divided by 47. Show the long division steps. | To divide 54712 by 47:
1. Determine how many times 47 goes into the leading digits of 54712.
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: 54712 / 47 = 1164 with a remainder of 4.
Proof: (1164 * 47)... |
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 |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[-3, -1]
B =
[2, -3]
[4, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*2) + (1*4) = 2
C12 = (A11 * B12) + (A12 * B22) = (-1*-3) + (1*4) = 7
C21 = (A21 * B11) + (A22 * B21) = (-3*2) + (-1*4) = -10
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (-1*4) = 5
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 16^2 = c^2
2. Calculate the squares: 49 + 256 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 14, 2, 10, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 14 + 2 + 10 + 16) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.0, 3.0, -9.0, -1.0, 5.0]
Squared Differences: [4.0, 9.0, 81.0, 1.0, 25.0]
Sum... |
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 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) |
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