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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... |
Compute the quotient and remainder of 78684 divided by 47. Show the long division steps. | To divide 78684 by 47:
1. Determine how many times 47 goes into the leading digits of 78684.
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: 78684 / 47 = 1674 with a remainder of 6.
Proof: (1674 * 47)... |
Multiply the following 2x2 matrices:
A =
[-2, -4]
[0, -3]
B =
[4, 5]
[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) = (-2*4) + (-4*4) = -24
C12 = (A11 * B12) + (A12 * B22) = (-2*5) + (-4*-5) = 10
C21 = (A21 * B11) + (A22 * B21) = (0*4) + (-3*4) = -12
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (-3*-5) = 15
Res... |
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 11/12 and 14/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 14, which is 84.
2. Convert 11/12 by multiplying top and bottom by 7: 77/84.
3. Convert 14/14 by multiplying top and bottom by 6: 84/84.
4. Add the numerators: 77 + 84 = 161.
5. The resulting fraction is 161/84.
6. Simplif... |
Find the roots of the quadratic equation: 2x^2 + 2x + -6 = 0 | 1. Identify coefficients: a=2, b=2, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -48 = 52.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √52) / 4.
5. Calculate roots: x1 = 1.3027756377319946, x2 = -2.302775637731995.
Final Answe... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 15, 11, 12, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 15 + 11 + 12 + 13) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 1.6, -2.4, -1.4, -0.4]
Squared Differences: [6.76, 2.56, 5.76, 1.96, 0.16]
... |
Find the roots of the quadratic equation: 5x^2 + 1x + 0 = 0 | 1. Identify coefficients: a=5, b=1, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - 0 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √1) / 10.
5. Calculate roots: x1 = 0.0, x2 = -0.2.
Final Answer: x = 0.0, -0.2 |
Compute the quotient and remainder of 48922 divided by 45. Show the long division steps. | To divide 48922 by 45:
1. Determine how many times 45 goes into the leading digits of 48922.
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: 48922 / 45 = 1087 with a remainder of 7.
Proof: (1087 * 45)... |
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 |
Find the roots of the quadratic equation: 7x^2 + 1x + -6 = 0 | 1. Identify coefficients: a=7, b=1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -168 = 169.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √169) / 14.
5. Calculate roots: x1 = 0.8571428571428571, x2 = -1.0.
Final Answer: x = 0.8... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Find the derivative of f(x) = (2x + 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 +... |
Compute the indefinite integral of f(x) = 7x^1. | To find the indefinite integral ∫ 7x^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: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 20, 9, 1, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 20 + 9 + 1 + 9) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, 11.2, 0.2, -7.8, 0.2]
Squared Differences: [14.44, 125.44, 0.04, 60.84, 0.04]
... |
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 |
Add the fractions 13/9 and 3/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 2, which is 18.
2. Convert 13/9 by multiplying top and bottom by 2: 26/18.
3. Convert 3/2 by multiplying top and bottom by 9: 27/18.
4. Add the numerators: 26 + 27 = 53.
5. The resulting fraction is 53/18.
Final Answer: 53/... |
Multiply the following 2x2 matrices:
A =
[-3, 0]
[4, 1]
B =
[-1, 0]
[-5, 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*-1) + (0*-5) = 3
C12 = (A11 * B12) + (A12 * B22) = (-3*0) + (0*3) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*-1) + (1*-5) = -9
C22 = (A21 * B12) + (A22 * B22) = (4*0) + (1*3) = 3
Result Mat... |
Compute the quotient and remainder of 94761 divided by 86. Show the long division steps. | To divide 94761 by 86:
1. Determine how many times 86 goes into the leading digits of 94761.
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: 94761 / 86 = 1101 with a remainder of 75.
Proof: (1101 * 86... |
Compute the quotient and remainder of 13721 divided by 49. Show the long division steps. | To divide 13721 by 49:
1. Determine how many times 49 goes into the leading digits of 13721.
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: 13721 / 49 = 280 with a remainder of 1.
Proof: (280 * 49) +... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 13. 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 + 13^2 = c^2
2. Calculate the squares: 256 + 169 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final A... |
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 =
[0, 0]
[4, 3]
B =
[0, -4]
[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) + (0*3) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*-4) + (0*1) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (3*3) = 9
C22 = (A21 * B12) + (A22 * B22) = (4*-4) + (3*1) = -13
Result Matrix... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 20^2 = c^2
2. Calculate the squares: 81 + 400 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 3^2 = c^2
2. Calculate the squares: 16 + 9 = c^2
3. Sum the squares: 25 = c^2
4. Take the square root of both sides: c = √25
Final Answer: ... |
Add the fractions 6/7 and 11/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 14, which is 14.
2. Convert 6/7 by multiplying top and bottom by 2: 12/14.
3. Convert 11/14 by multiplying top and bottom by 1: 11/14.
4. Add the numerators: 12 + 11 = 23.
5. The resulting fraction is 23/14.
Final Answer: 2... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Evaluate 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 = 9 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: 9^2 + 19^2 = c^2
2. Calculate the squares: 81 + 361 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
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 ... |
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 |
Add the fractions 13/3 and 5/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 7, which is 21.
2. Convert 13/3 by multiplying top and bottom by 7: 91/21.
3. Convert 5/7 by multiplying top and bottom by 3: 15/21.
4. Add the numerators: 91 + 15 = 106.
5. The resulting fraction is 106/21.
Final Answer: 1... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 17, 13, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 17 + 13 + 9 + 10) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.4, 3.4, -0.6, -4.6, -3.6]
Squared Differences: [29.16, 11.56, 0.36, 21.16, 12.96... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) 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) |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 8^2 = c^2
2. Calculate the squares: 400 + 64 = c^2
3. Sum the squares: 464 = c^2
4. Take the square root of both sides: c = √464
Final Ans... |
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) |
Add the fractions 1/6 and 12/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 15, which is 30.
2. Convert 1/6 by multiplying top and bottom by 5: 5/30.
3. Convert 12/15 by multiplying top and bottom by 2: 24/30.
4. Add the numerators: 5 + 24 = 29.
5. The resulting fraction is 29/30.
Final Answer: 29/... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[0, 0]
[4, 3]
B =
[-3, -3]
[-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) = (0*-3) + (0*-4) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*-3) + (0*-3) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*-3) + (3*-4) = -24
C22 = (A21 * B12) + (A22 * B22) = (4*-3) + (3*-3) = -21
Resul... |
Compute the quotient and remainder of 93597 divided by 60. Show the long division steps. | To divide 93597 by 60:
1. Determine how many times 60 goes into the leading digits of 93597.
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: 93597 / 60 = 1559 with a remainder of 57.
Proof: (1559 * 60... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 16, 20, 4, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 16 + 20 + 4 + 3) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, 5.8, 9.8, -6.2, -7.2]
Squared Differences: [4.84, 33.64, 96.04, 38.44, 51.84]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 10, 3, 3, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 10 + 3 + 3 + 14) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, 2.6, -4.4, -4.4, 6.6]
Squared Differences: [0.16, 6.76, 19.36, 19.36, 43.56]
... |
Compute the quotient and remainder of 33314 divided by 41. Show the long division steps. | To divide 33314 by 41:
1. Determine how many times 41 goes into the leading digits of 33314.
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: 33314 / 41 = 812 with a remainder of 22.
Proof: (812 * 41) ... |
Find the roots of the quadratic equation: 8x^2 + -7x + 0 = 0 | 1. Identify coefficients: a=8, b=-7, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 0 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √49) / 16.
5. Calculate roots: x1 = 0.875, x2 = 0.0.
Final Answer: x = 0.875, 0.0 |
Multiply the following 2x2 matrices:
A =
[-5, -5]
[-5, -5]
B =
[4, -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) = (-5*4) + (-5*3) = -35
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (-5*-2) = 25
C21 = (A21 * B11) + (A22 * B21) = (-5*4) + (-5*3) = -35
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (-5*-2) = 25
... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 17, 2, 16, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 17 + 2 + 16 + 10) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, 6.4, -8.6, 5.4, -0.6]
Squared Differences: [6.76, 40.96, 73.96, 29.16, 0.36]
... |
Multiply the following 2x2 matrices:
A =
[4, 1]
[-3, 3]
B =
[4, -5]
[-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) = (4*4) + (1*-5) = 11
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (1*1) = -19
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (3*-5) = -27
C22 = (A21 * B12) + (A22 * B22) = (-3*-5) + (3*1) = 18
Resul... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 14^2 = c^2
2. Calculate the squares: 25 + 196 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 7, 18, 17, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 7 + 18 + 17 + 6) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.8, -4.8, 6.2, 5.2, -5.8]
Squared Differences: [0.64, 23.04, 38.44, 27.04, 33.64]... |
Add the fractions 9/5 and 9/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 8, which is 40.
2. Convert 9/5 by multiplying top and bottom by 8: 72/40.
3. Convert 9/8 by multiplying top and bottom by 5: 45/40.
4. Add the numerators: 72 + 45 = 117.
5. The resulting fraction is 117/40.
Final Answer: 11... |
Multiply the following 2x2 matrices:
A =
[3, -3]
[-5, 4]
B =
[3, 2]
[1, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*3) + (-3*1) = 6
C12 = (A11 * B12) + (A12 * B22) = (3*2) + (-3*-3) = 15
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (4*1) = -11
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (4*-3) = -22
Result... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 8^2 = c^2
2. Calculate the squares: 196 + 64 = c^2
3. Sum the squares: 260 = c^2
4. Take the square root of both sides: c = √260
Final Ans... |
Compute the quotient and remainder of 93482 divided by 81. Show the long division steps. | To divide 93482 by 81:
1. Determine how many times 81 goes into the leading digits of 93482.
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: 93482 / 81 = 1154 with a remainder of 8.
Proof: (1154 * 81)... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 2, 20, 11, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 2 + 20 + 11 + 11) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.0, -7.0, 11.0, 2.0, 2.0]
Squared Differences: [64.0, 49.0, 121.0, 4.0, 4.0]
Su... |
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 |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 92870 divided by 95. Show the long division steps. | To divide 92870 by 95:
1. Determine how many times 95 goes into the leading digits of 92870.
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: 92870 / 95 = 977 with a remainder of 55.
Proof: (977 * 95) ... |
Find the roots of the quadratic equation: 5x^2 + 10x + 4 = 0 | 1. Identify coefficients: a=5, b=10, c=4.
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) / 10.
5. Calculate roots: x1 = -0.552786404500042, x2 = -1.4472135954999579.
Final A... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 5x^2 + -6x + 10 = 0 | 1. Identify coefficients: a=5, b=-6, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(5)(10) = 36 - 200 = -164.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √164i) / 10.
Final Answer: x = 0.6 ± 1.2806248474865698i |
Add the fractions 1/12 and 6/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 15, which is 60.
2. Convert 1/12 by multiplying top and bottom by 5: 5/60.
3. Convert 6/15 by multiplying top and bottom by 4: 24/60.
4. Add the numerators: 5 + 24 = 29.
5. The resulting fraction is 29/60.
Final Answer: 29... |
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 |
Compute the quotient and remainder of 97822 divided by 76. Show the long division steps. | To divide 97822 by 76:
1. Determine how many times 76 goes into the leading digits of 97822.
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: 97822 / 76 = 1287 with a remainder of 10.
Proof: (1287 * 76... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[1, -3]
B =
[1, 2]
[-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) = (-1*1) + (3*-2) = -7
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (3*0) = -2
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (-3*-2) = 7
C22 = (A21 * B12) + (A22 * B22) = (1*2) + (-3*0) = 2
Result Ma... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 11, 1, 4, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 11 + 1 + 4 + 11) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.0, 2.0, -8.0, -5.0, 2.0]
Squared Differences: [81.0, 4.0, 64.0, 25.0, 4.0]
Sum ... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
Multiply the following 2x2 matrices:
A =
[5, -1]
[4, 0]
B =
[0, 0]
[-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) = (5*0) + (-1*-3) = 3
C12 = (A11 * B12) + (A12 * B22) = (5*0) + (-1*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (0*-3) = 0
C22 = (A21 * B12) + (A22 * B22) = (4*0) + (0*1) = 0
Result Matri... |
Multiply the following 2x2 matrices:
A =
[-2, -1]
[-4, -3]
B =
[0, 0]
[-5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*0) + (-1*-5) = 5
C12 = (A11 * B12) + (A12 * B22) = (-2*0) + (-1*-2) = 2
C21 = (A21 * B11) + (A22 * B21) = (-4*0) + (-3*-5) = 15
C22 = (A21 * B12) + (A22 * B22) = (-4*0) + (-3*-2) = 6
Resu... |
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) |
Find the roots of the quadratic equation: 1x^2 + 5x + 0 = 0 | 1. Identify coefficients: a=1, b=5, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 0 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √25) / 2.
5. Calculate roots: x1 = 0.0, x2 = -5.0.
Final Answer: x = 0.0, -5.0 |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[-1, -1]
B =
[5, 2]
[-4, -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) = (-2*5) + (2*-4) = -18
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (2*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (-1*-4) = -1
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (-1*-2) = 0
Res... |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 13. 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 + 13^2 = c^2
2. Calculate the squares: 289 + 169 = c^2
3. Sum the squares: 458 = c^2
4. Take the square root of both sides: c = √458
Final A... |
Find the roots of the quadratic equation: 10x^2 + -3x + -4 = 0 | 1. Identify coefficients: a=10, b=-3, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -160 = 169.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √169) / 20.
5. Calculate roots: x1 = 0.8, x2 = -0.5.
Final Answer: x = 0.8, -0.5 |
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 ... |
Find the roots of the quadratic equation: 7x^2 + 1x + -2 = 0 | 1. Identify coefficients: a=7, b=1, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -56 = 57.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √57) / 14.
5. Calculate roots: x1 = 0.46784531680505353, x2 = -0.6107024596621964.
Final An... |
Compute the quotient and remainder of 24333 divided by 93. Show the long division steps. | To divide 24333 by 93:
1. Determine how many times 93 goes into the leading digits of 24333.
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: 24333 / 93 = 261 with a remainder of 60.
Proof: (261 * 93) ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 9^2 = c^2
2. Calculate the squares: 256 + 81 = c^2
3. Sum the squares: 337 = c^2
4. Take the square root of both sides: c = √337
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 |
Find the roots of the quadratic equation: 7x^2 + -3x + 7 = 0 | 1. Identify coefficients: a=7, b=-3, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(7)(7) = 9 - 196 = -187.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √187i) / 14.
Final Answer: x = 0.21428571428571427 ± 0.9767710236555246i |
Compute the quotient and remainder of 15044 divided by 84. Show the long division steps. | To divide 15044 by 84:
1. Determine how many times 84 goes into the leading digits of 15044.
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: 15044 / 84 = 179 with a remainder of 8.
Proof: (179 * 84) +... |
Compute the quotient and remainder of 44238 divided by 84. Show the long division steps. | To divide 44238 by 84:
1. Determine how many times 84 goes into the leading digits of 44238.
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: 44238 / 84 = 526 with a remainder of 54.
Proof: (526 * 84) ... |
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 |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 7, 19, 20, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 7 + 19 + 20 + 5) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.0, -6.0, 6.0, 7.0, -8.0]
Squared Differences: [1.0, 36.0, 36.0, 49.0, 64.0]
Su... |
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) |
Compute the quotient and remainder of 16382 divided by 28. Show the long division steps. | To divide 16382 by 28:
1. Determine how many times 28 goes into the leading digits of 16382.
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: 16382 / 28 = 585 with a remainder of 2.
Proof: (585 * 28) +... |
Multiply the following 2x2 matrices:
A =
[1, -5]
[-1, 3]
B =
[-5, -3]
[-4, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-5) + (-5*-4) = 15
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (-5*-2) = 7
C21 = (A21 * B11) + (A22 * B21) = (-1*-5) + (3*-4) = -7
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (3*-2) = -3
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 9, 1, 20, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 9 + 1 + 20 + 10) / 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, -2.4, -10.4, 8.6, -1.4]
Squared Differences: [31.36, 5.76, 108.16, 73.96, 1.96... |
Compute the quotient and remainder of 76257 divided by 66. Show the long division steps. | To divide 76257 by 66:
1. Determine how many times 66 goes into the leading digits of 76257.
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: 76257 / 66 = 1155 with a remainder of 27.
Proof: (1155 * 66... |
Find the roots of the quadratic equation: 1x^2 + -6x + 8 = 0 | 1. Identify coefficients: a=1, b=-6, c=8.
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) / 2.
5. Calculate roots: x1 = 4.0, x2 = 2.0.
Final Answer: x = 4.0, 2.0 |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 8, 7, 1, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 8 + 7 + 1 + 20) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, -0.4, -1.4, -7.4, 11.6]
Squared Differences: [5.76, 0.16, 1.96, 54.76, 134.56]
... |
Add the fractions 10/9 and 8/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 6, which is 18.
2. Convert 10/9 by multiplying top and bottom by 2: 20/18.
3. Convert 8/6 by multiplying top and bottom by 3: 24/18.
4. Add the numerators: 20 + 24 = 44.
5. The resulting fraction is 44/18.
6. Simplify by di... |
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 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 = 17 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: 17^2 + 8^2 = c^2
2. Calculate the squares: 289 + 64 = c^2
3. Sum the squares: 353 = c^2
4. Take the square root of both sides: c = √353
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 =
[-3, 1]
[4, 1]
B =
[-1, -5]
[0, 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*-1) + (1*0) = 3
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (1*3) = 18
C21 = (A21 * B11) + (A22 * B21) = (4*-1) + (1*0) = -4
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (1*3) = -17
Result ... |
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