prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
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 |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 9^2 = c^2
2. Calculate the squares: 9 + 81 = c^2
3. Sum the squares: 90 = c^2
4. Take the square root of both sides: c = √90
Final Answer: ... |
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 ... |
Add the fractions 8/4 and 7/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 14, which is 28.
2. Convert 8/4 by multiplying top and bottom by 7: 56/28.
3. Convert 7/14 by multiplying top and bottom by 2: 14/28.
4. Add the numerators: 56 + 14 = 70.
5. The resulting fraction is 70/28.
6. Simplify by d... |
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 4/15 and 14/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 6, which is 30.
2. Convert 4/15 by multiplying top and bottom by 2: 8/30.
3. Convert 14/6 by multiplying top and bottom by 5: 70/30.
4. Add the numerators: 8 + 70 = 78.
5. The resulting fraction is 78/30.
6. Simplify by di... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 9^2 = c^2
2. Calculate the squares: 196 + 81 = c^2
3. Sum the squares: 277 = c^2
4. Take the square root of both sides: c = √277
Final Ans... |
Compute the quotient and remainder of 17478 divided by 66. Show the long division steps. | To divide 17478 by 66:
1. Determine how many times 66 goes into the leading digits of 17478.
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: 17478 / 66 = 264 with a remainder of 54.
Proof: (264 * 66) ... |
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 |
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) |
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 |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Compute the quotient and remainder of 73151 divided by 70. Show the long division steps. | To divide 73151 by 70:
1. Determine how many times 70 goes into the leading digits of 73151.
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: 73151 / 70 = 1045 with a remainder of 1.
Proof: (1045 * 70)... |
Add the fractions 5/11 and 6/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 5, which is 55.
2. Convert 5/11 by multiplying top and bottom by 5: 25/55.
3. Convert 6/5 by multiplying top and bottom by 11: 66/55.
4. Add the numerators: 25 + 66 = 91.
5. The resulting fraction is 91/55.
Final Answer: 9... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 8^2 = c^2
2. Calculate the squares: 36 + 64 = c^2
3. Sum the squares: 100 = c^2
4. Take the square root of both sides: c = √100
Final Answe... |
Add the fractions 8/6 and 9/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 5, which is 30.
2. Convert 8/6 by multiplying top and bottom by 5: 40/30.
3. Convert 9/5 by multiplying top and bottom by 6: 54/30.
4. Add the numerators: 40 + 54 = 94.
5. The resulting fraction is 94/30.
6. Simplify by div... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 12, 20, 10, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 12 + 20 + 10 + 4) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, 2.0, 10.0, 0.0, -6.0]
Squared Differences: [36.0, 4.0, 100.0, 0.0, 36.0]
S... |
Add the fractions 11/2 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 6, which is 6.
2. Convert 11/2 by multiplying top and bottom by 3: 33/6.
3. Convert 3/6 by multiplying top and bottom by 1: 3/6.
4. Add the numerators: 33 + 3 = 36.
5. The resulting fraction is 36/6.
6. Simplify by dividing... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 7, 19, 6, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 7 + 19 + 6 + 3) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -1.2, 10.8, -2.2, -5.2]
Squared Differences: [4.84, 1.44, 116.64, 4.84, 27.04]
... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 3^2 = c^2
2. Calculate the squares: 256 + 9 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Find the roots of the quadratic equation: 1x^2 + 4x + 1 = 0 | 1. Identify coefficients: a=1, b=4, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - 4 = 12.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √12) / 2.
5. Calculate roots: x1 = -0.2679491924311228, x2 = -3.732050807568877.
Final Answer... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 10, 20, 15, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 10 + 20 + 15 + 17) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.8, -3.8, 6.2, 1.2, 3.2]
Squared Differences: [46.24, 14.44, 38.44, 1.44, 10.24]... |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[-2, 2]
B =
[1, 4]
[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) = (-3*1) + (5*2) = 7
C12 = (A11 * B12) + (A12 * B22) = (-3*4) + (5*2) = -2
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (2*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (-2*4) + (2*2) = -4
Result Matr... |
Add the fractions 14/4 and 14/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 8, which is 8.
2. Convert 14/4 by multiplying top and bottom by 2: 28/8.
3. Convert 14/8 by multiplying top and bottom by 1: 14/8.
4. Add the numerators: 28 + 14 = 42.
5. The resulting fraction is 42/8.
6. Simplify by divid... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 15^2 = c^2
2. Calculate the squares: 361 + 225 = c^2
3. Sum the squares: 586 = c^2
4. Take the square root of both sides: c = √586
Final A... |
Compute the quotient and remainder of 41672 divided by 60. Show the long division steps. | To divide 41672 by 60:
1. Determine how many times 60 goes into the leading digits of 41672.
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: 41672 / 60 = 694 with a remainder of 32.
Proof: (694 * 60) ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 7^2 = c^2
2. Calculate the squares: 256 + 49 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 16. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 16^2 = c^2
2. Calculate the squares: 289 + 256 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[-3, 4]
B =
[4, 1]
[3, -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*4) + (-2*3) = -22
C12 = (A11 * B12) + (A12 * B22) = (-4*1) + (-2*-5) = 6
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (4*3) = 0
C22 = (A21 * B12) + (A22 * B22) = (-3*1) + (4*-5) = -23
Resul... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 18, 18, 19, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 18 + 18 + 19 + 16) / 5 = 88 / 5 = 17.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.6, 0.4, 0.4, 1.4, -1.6]
Squared Differences: [0.36, 0.16, 0.16, 1.96, 2.56]
... |
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 |
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 |
Compute the quotient and remainder of 11070 divided by 40. Show the long division steps. | To divide 11070 by 40:
1. Determine how many times 40 goes into the leading digits of 11070.
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: 11070 / 40 = 276 with a remainder of 30.
Proof: (276 * 40) ... |
Find the roots of the quadratic equation: 8x^2 + -5x + 6 = 0 | 1. Identify coefficients: a=8, b=-5, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(8)(6) = 25 - 192 = -167.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √167i) / 16.
Final Answer: x = 0.3125 ± 0.8076779989575054i |
Compute the quotient and remainder of 71134 divided by 59. Show the long division steps. | To divide 71134 by 59:
1. Determine how many times 59 goes into the leading digits of 71134.
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: 71134 / 59 = 1205 with a remainder of 39.
Proof: (1205 * 59... |
Compute the quotient and remainder of 8191 divided by 38. Show the long division steps. | To divide 8191 by 38:
1. Determine how many times 38 goes into the leading digits of 8191.
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: 8191 / 38 = 215 with a remainder of 21.
Proof: (215 * 38) + 2... |
Add the fractions 11/11 and 1/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 5, which is 55.
2. Convert 11/11 by multiplying top and bottom by 5: 55/55.
3. Convert 1/5 by multiplying top and bottom by 11: 11/55.
4. Add the numerators: 55 + 11 = 66.
5. The resulting fraction is 66/55.
6. Simplify by... |
Compute the quotient and remainder of 72815 divided by 8. Show the long division steps. | To divide 72815 by 8:
1. Determine how many times 8 goes into the leading digits of 72815.
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: 72815 / 8 = 9101 with a remainder of 7.
Proof: (9101 * 8) + 7... |
Find the roots of the quadratic equation: 3x^2 + -3x + -3 = 0 | 1. Identify coefficients: a=3, b=-3, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -36 = 45.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √45) / 6.
5. Calculate roots: x1 = 1.618033988749895, x2 = -0.6180339887498949.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[4, -3]
[4, 1]
B =
[0, 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) = (4*0) + (-3*1) = -3
C12 = (A11 * B12) + (A12 * B22) = (4*4) + (-3*-5) = 31
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (1*1) = 1
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (1*-5) = 11
Result Mat... |
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 |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^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: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Find the roots of the quadratic equation: 2x^2 + 3x + 3 = 0 | 1. Identify coefficients: a=2, b=3, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(2)(3) = 9 - 24 = -15.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √15i) / 4.
Final Answer: x = -0.75 ± 0.9682458365518543i |
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 |
Find the roots of the quadratic equation: 9x^2 + 1x + -10 = 0 | 1. Identify coefficients: a=9, b=1, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -360 = 361.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √361) / 18.
5. Calculate roots: x1 = 1.0, x2 = -1.1111111111111112.
Final Answer: x = 1.... |
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 89183 divided by 87. Show the long division steps. | To divide 89183 by 87:
1. Determine how many times 87 goes into the leading digits of 89183.
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: 89183 / 87 = 1025 with a remainder of 8.
Proof: (1025 * 87)... |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 18. 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 + 18^2 = c^2
2. Calculate the squares: 324 + 324 = c^2
3. Sum the squares: 648 = c^2
4. Take the square root of both sides: c = √648
Final A... |
Find the roots of the quadratic equation: 4x^2 + 6x + 7 = 0 | 1. Identify coefficients: a=4, b=6, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(4)(7) = 36 - 112 = -76.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √76i) / 8.
Final Answer: x = -0.75 ± 1.0897247358851685i |
Add the fractions 7/3 and 11/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 11, which is 33.
2. Convert 7/3 by multiplying top and bottom by 11: 77/33.
3. Convert 11/11 by multiplying top and bottom by 3: 33/33.
4. Add the numerators: 77 + 33 = 110.
5. The resulting fraction is 110/33.
6. Simplify ... |
Find the roots of the quadratic equation: 7x^2 + 2x + -3 = 0 | 1. Identify coefficients: a=7, b=2, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -84 = 88.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √88) / 14.
5. Calculate roots: x1 = 0.5272022514033471, x2 = -0.8129165371176328.
Final Ans... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[4, 1]
B =
[3, -1]
[-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) = (2*3) + (3*-2) = 0
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (3*5) = 13
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (1*-2) = 10
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (1*5) = 1
Result Matr... |
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 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 2, 11, 6, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 2 + 11 + 6 + 10) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, -6.4, 2.6, -2.4, 1.6]
Squared Differences: [21.16, 40.96, 6.76, 5.76, 2.56]
... |
Multiply the following 2x2 matrices:
A =
[-1, -3]
[3, 2]
B =
[2, 5]
[-1, 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*2) + (-3*-1) = 1
C12 = (A11 * B12) + (A12 * B22) = (-1*5) + (-3*1) = -8
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (2*-1) = 4
C22 = (A21 * B12) + (A22 * B22) = (3*5) + (2*1) = 17
Result Ma... |
Add the fractions 11/15 and 10/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 5, which is 15.
2. Convert 11/15 by multiplying top and bottom by 1: 11/15.
3. Convert 10/5 by multiplying top and bottom by 3: 30/15.
4. Add the numerators: 11 + 30 = 41.
5. The resulting fraction is 41/15.
Final Answer: ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 2, 18, 8, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 2 + 18 + 8 + 20) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.8, -7.8, 8.2, -1.8, 10.2]
Squared Differences: [77.44, 60.84, 67.24, 3.24, 104.04]... |
Find the roots of the quadratic equation: 4x^2 + -4x + -1 = 0 | 1. Identify coefficients: a=4, b=-4, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -16 = 32.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √32) / 8.
5. Calculate roots: x1 = 1.2071067811865475, x2 = -0.20710678118654757.
Final An... |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[5, -4]
[-4, -1]
B =
[1, 4]
[3, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*1) + (-4*3) = -7
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (-4*-3) = 32
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (-1*3) = -7
C22 = (A21 * B12) + (A22 * B22) = (-4*4) + (-1*-3) = -13
Resu... |
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 ... |
Multiply the following 2x2 matrices:
A =
[1, -5]
[-1, -4]
B =
[-3, -4]
[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) = (1*-3) + (-5*0) = -3
C12 = (A11 * B12) + (A12 * B22) = (1*-4) + (-5*3) = -19
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (-4*0) = 3
C22 = (A21 * B12) + (A22 * B22) = (-1*-4) + (-4*3) = -8
Res... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 16^2 = c^2
2. Calculate the squares: 361 + 256 = c^2
3. Sum the squares: 617 = c^2
4. Take the square root of both sides: c = √617
Final A... |
Find the roots of the quadratic equation: 3x^2 + -2x + 7 = 0 | 1. Identify coefficients: a=3, b=-2, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(3)(7) = 4 - 84 = -80.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √80i) / 6.
Final Answer: x = 0.3333333333333333 ± 1.4907119849998598i |
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 ... |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 46897 divided by 30. Show the long division steps. | To divide 46897 by 30:
1. Determine how many times 30 goes into the leading digits of 46897.
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: 46897 / 30 = 1563 with a remainder of 7.
Proof: (1563 * 30)... |
Add the fractions 9/11 and 6/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 9/11 by multiplying top and bottom by 7: 63/77.
3. Convert 6/7 by multiplying top and bottom by 11: 66/77.
4. Add the numerators: 63 + 66 = 129.
5. The resulting fraction is 129/77.
Final Answer:... |
Add the fractions 9/8 and 8/14. 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 9/8 by multiplying top and bottom by 7: 63/56.
3. Convert 8/14 by multiplying top and bottom by 4: 32/56.
4. Add the numerators: 63 + 32 = 95.
5. The resulting fraction is 95/56.
Final Answer: 95... |
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 |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 3x^3. | To find the indefinite integral ∫ 3x^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: 3 / 4 = 3/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^3 dx = 3/4x^4 + 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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 3, 8, 5, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 3 + 8 + 5 + 9) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, -4.4, 0.6, -2.4, 1.6]
Squared Differences: [21.16, 19.36, 0.36, 5.76, 2.56]
Su... |
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 |
Find the roots of the quadratic equation: 3x^2 + 3x + 1 = 0 | 1. Identify coefficients: a=3, b=3, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(3)(1) = 9 - 12 = -3.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √3i) / 6.
Final Answer: x = -0.5 ± 0.28867513459481287i |
Add the fractions 11/14 and 10/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 12, which is 84.
2. Convert 11/14 by multiplying top and bottom by 6: 66/84.
3. Convert 10/12 by multiplying top and bottom by 7: 70/84.
4. Add the numerators: 66 + 70 = 136.
5. The resulting fraction is 136/84.
6. Simplif... |
Add the fractions 4/8 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 4/8 by multiplying top and bottom by 1: 4/8.
3. Convert 3/8 by multiplying top and bottom by 1: 3/8.
4. Add the numerators: 4 + 3 = 7.
5. The resulting fraction is 7/8.
Final Answer: 7/8 |
Find the roots of the quadratic equation: 3x^2 + 3x + 3 = 0 | 1. Identify coefficients: a=3, b=3, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(3)(3) = 9 - 36 = -27.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √27i) / 6.
Final Answer: x = -0.5 ± 0.8660254037844387i |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 11, 13, 2, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 11 + 13 + 2 + 17) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, 1.0, 3.0, -8.0, 7.0]
Squared Differences: [9.0, 1.0, 9.0, 64.0, 49.0]
Sum ... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 7^2 = c^2
2. Calculate the squares: 324 + 49 = c^2
3. Sum the squares: 373 = c^2
4. Take the square root of both sides: c = √373
Final Ans... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 15267 divided by 91. Show the long division steps. | To divide 15267 by 91:
1. Determine how many times 91 goes into the leading digits of 15267.
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: 15267 / 91 = 167 with a remainder of 70.
Proof: (167 * 91) ... |
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 51199 divided by 51. Show the long division steps. | To divide 51199 by 51:
1. Determine how many times 51 goes into the leading digits of 51199.
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: 51199 / 51 = 1003 with a remainder of 46.
Proof: (1003 * 51... |
Multiply the following 2x2 matrices:
A =
[3, 2]
[1, -1]
B =
[-2, -4]
[-5, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*-2) + (2*-5) = -16
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (2*-5) = -22
C21 = (A21 * B11) + (A22 * B21) = (1*-2) + (-1*-5) = 3
C22 = (A21 * B12) + (A22 * B22) = (1*-4) + (-1*-5) = 1
Res... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
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 ... |
Add the fractions 15/4 and 12/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 11, which is 44.
2. Convert 15/4 by multiplying top and bottom by 11: 165/44.
3. Convert 12/11 by multiplying top and bottom by 4: 48/44.
4. Add the numerators: 165 + 48 = 213.
5. The resulting fraction is 213/44.
Final Ans... |
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) |
Add the fractions 13/5 and 5/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 6, which is 30.
2. Convert 13/5 by multiplying top and bottom by 6: 78/30.
3. Convert 5/6 by multiplying top and bottom by 5: 25/30.
4. Add the numerators: 78 + 25 = 103.
5. The resulting fraction is 103/30.
Final Answer: 1... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.