prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Multiply the following 2x2 matrices:
A =
[1, 3]
[5, -4]
B =
[-5, 1]
[-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) = (1*-5) + (3*-3) = -14
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (3*-4) = -11
C21 = (A21 * B11) + (A22 * B21) = (5*-5) + (-4*-3) = -13
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (-4*-4) = 21
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 16, 20, 17, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 16 + 20 + 17 + 5) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 2.2, 6.2, 3.2, -8.8]
Squared Differences: [7.84, 4.84, 38.44, 10.24, 77.44]
... |
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 quotient and remainder of 76797 divided by 36. Show the long division steps. | To divide 76797 by 36:
1. Determine how many times 36 goes into the leading digits of 76797.
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: 76797 / 36 = 2133 with a remainder of 9.
Proof: (2133 * 36)... |
Compute the quotient and remainder of 65119 divided by 32. Show the long division steps. | To divide 65119 by 32:
1. Determine how many times 32 goes into the leading digits of 65119.
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: 65119 / 32 = 2034 with a remainder of 31.
Proof: (2034 * 32... |
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 |
Add the fractions 13/14 and 10/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 4, which is 28.
2. Convert 13/14 by multiplying top and bottom by 2: 26/28.
3. Convert 10/4 by multiplying top and bottom by 7: 70/28.
4. Add the numerators: 26 + 70 = 96.
5. The resulting fraction is 96/28.
6. Simplify by... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Add the fractions 14/7 and 14/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 5, which is 35.
2. Convert 14/7 by multiplying top and bottom by 5: 70/35.
3. Convert 14/5 by multiplying top and bottom by 7: 98/35.
4. Add the numerators: 70 + 98 = 168.
5. The resulting fraction is 168/35.
6. Simplify by... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 7^2 = c^2
2. Calculate the squares: 400 + 49 = c^2
3. Sum the squares: 449 = c^2
4. Take the square root of both sides: c = √449
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + -9x + -1 = 0 | 1. Identify coefficients: a=7, b=-9, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -28 = 109.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √109) / 14.
5. Calculate roots: x1 = 1.3885933220650395, x2 = -0.10287903635075361.
Final... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 4, 8, 10, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 4 + 8 + 10 + 9) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.2, -5.8, -1.8, 0.2, -0.8]
Squared Differences: [67.24, 33.64, 3.24, 0.04, 0.64]
... |
Add the fractions 13/4 and 6/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 12, which is 12.
2. Convert 13/4 by multiplying top and bottom by 3: 39/12.
3. Convert 6/12 by multiplying top and bottom by 1: 6/12.
4. Add the numerators: 39 + 6 = 45.
5. The resulting fraction is 45/12.
6. Simplify by di... |
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 +... |
Find the roots of the quadratic equation: 2x^2 + 1x + -5 = 0 | 1. Identify coefficients: a=2, b=1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -40 = 41.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √41) / 4.
5. Calculate roots: x1 = 1.3507810593582121, x2 = -1.8507810593582121.
Final Answ... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 15^2 = c^2
2. Calculate the squares: 64 + 225 = c^2
3. Sum the squares: 289 = c^2
4. Take the square root of both sides: c = √289
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 16^2 = c^2
2. Calculate the squares: 16 + 256 = c^2
3. Sum the squares: 272 = c^2
4. Take the square root of both sides: c = √272
Final Ans... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 46314 divided by 70. Show the long division steps. | To divide 46314 by 70:
1. Determine how many times 70 goes into the leading digits of 46314.
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: 46314 / 70 = 661 with a remainder of 44.
Proof: (661 * 70) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 2, 16, 6, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 2 + 16 + 6 + 13) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, -7.2, 6.8, -3.2, 3.8]
Squared Differences: [0.04, 51.84, 46.24, 10.24, 14.44]
... |
Find the roots of the quadratic equation: 1x^2 + -1x + -10 = 0 | 1. Identify coefficients: a=1, b=-1, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -40 = 41.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √41) / 2.
5. Calculate roots: x1 = 3.7015621187164243, x2 = -2.7015621187164243.
Final Ans... |
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 4/4 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 3, which is 12.
2. Convert 4/4 by multiplying top and bottom by 3: 12/12.
3. Convert 14/3 by multiplying top and bottom by 4: 56/12.
4. Add the numerators: 12 + 56 = 68.
5. The resulting fraction is 68/12.
6. Simplify by di... |
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 14772 divided by 66. Show the long division steps. | To divide 14772 by 66:
1. Determine how many times 66 goes into the leading digits of 14772.
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: 14772 / 66 = 223 with a remainder of 54.
Proof: (223 * 66) ... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[1, -1]
B =
[-4, -3]
[-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) = (1*-4) + (-2*-1) = -2
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (-2*5) = -13
C21 = (A21 * B11) + (A22 * B21) = (1*-4) + (-1*-1) = -3
C22 = (A21 * B12) + (A22 * B22) = (1*-3) + (-1*5) = -8
Re... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 11, 11, 6, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 11 + 11 + 6 + 17) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.0, -1.0, -1.0, -6.0, 5.0]
Squared Differences: [9.0, 1.0, 1.0, 36.0, 25.0]
Su... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 20, 9, 9, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 20 + 9 + 9 + 16) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, 8.4, -2.6, -2.6, 4.4]
Squared Differences: [57.76, 70.56, 6.76, 6.76, 19.36]
... |
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 ... |
Compute the indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^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: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 49 + 25 = c^2
3. Sum the squares: 74 = c^2
4. Take the square root of both sides: c = √74
Final Answer:... |
Find the roots of the quadratic equation: 10x^2 + -1x + 6 = 0 | 1. Identify coefficients: a=10, b=-1, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(10)(6) = 1 - 240 = -239.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √239i) / 20.
Final Answer: x = 0.05 ± 0.7729812416870153i |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 7, 16, 14, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 7 + 16 + 14 + 19) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.6, -5.6, 3.4, 1.4, 6.4]
Squared Differences: [31.36, 31.36, 11.56, 1.96, 40.96]
... |
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) |
Add the fractions 4/10 and 7/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 9, which is 90.
2. Convert 4/10 by multiplying top and bottom by 9: 36/90.
3. Convert 7/9 by multiplying top and bottom by 10: 70/90.
4. Add the numerators: 36 + 70 = 106.
5. The resulting fraction is 106/90.
6. Simplify b... |
Find the roots of the quadratic equation: 8x^2 + 9x + 1 = 0 | 1. Identify coefficients: a=8, b=9, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 32 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √49) / 16.
5. Calculate roots: x1 = -0.125, x2 = -1.0.
Final Answer: x = -0.125, -1.0 |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 3^2 = c^2
2. Calculate the squares: 100 + 9 = c^2
3. Sum the squares: 109 = c^2
4. Take the square root of both sides: c = √109
Final Answ... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-2, -1]
B =
[1, 2]
[-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) = (2*1) + (3*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (2*2) + (3*1) = 7
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (-1*-1) = -1
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (-1*1) = -5
Result M... |
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 |
Compute the quotient and remainder of 31526 divided by 16. Show the long division steps. | To divide 31526 by 16:
1. Determine how many times 16 goes into the leading digits of 31526.
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: 31526 / 16 = 1970 with a remainder of 6.
Proof: (1970 * 16)... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 14^2 = c^2
2. Calculate the squares: 9 + 196 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Answ... |
Find the roots of the quadratic equation: 4x^2 + 6x + -10 = 0 | 1. Identify coefficients: a=4, b=6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -160 = 196.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √196) / 8.
5. Calculate roots: x1 = 1.0, x2 = -2.5.
Final Answer: x = 1.0, -2.5 |
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 quotient and remainder of 74732 divided by 12. Show the long division steps. | To divide 74732 by 12:
1. Determine how many times 12 goes into the leading digits of 74732.
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: 74732 / 12 = 6227 with a remainder of 8.
Proof: (6227 * 12)... |
Multiply the following 2x2 matrices:
A =
[4, -4]
[1, 5]
B =
[5, -3]
[-4, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*5) + (-4*-4) = 36
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (-4*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (5*-4) = -15
C22 = (A21 * B12) + (A22 * B22) = (1*-3) + (5*0) = -3
Resul... |
Compute the quotient and remainder of 58744 divided by 54. Show the long division steps. | To divide 58744 by 54:
1. Determine how many times 54 goes into the leading digits of 58744.
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: 58744 / 54 = 1087 with a remainder of 46.
Proof: (1087 * 54... |
Find the roots of the quadratic equation: 1x^2 + -1x + -5 = 0 | 1. Identify coefficients: a=1, b=-1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -20 = 21.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √21) / 2.
5. Calculate roots: x1 = 2.79128784747792, x2 = -1.79128784747792.
Final Answer: ... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 18^2 = c^2
2. Calculate the squares: 100 + 324 = c^2
3. Sum the squares: 424 = c^2
4. Take the square root of both sides: c = √424
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 1, 9, 12, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 1 + 9 + 12 + 13) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.0, -10.0, -2.0, 1.0, 2.0]
Squared Differences: [81.0, 100.0, 4.0, 1.0, 4.0]
Su... |
Compute the quotient and remainder of 3008 divided by 20. Show the long division steps. | To divide 3008 by 20:
1. Determine how many times 20 goes into the leading digits of 3008.
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: 3008 / 20 = 150 with a remainder of 8.
Proof: (150 * 20) + 8 ... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 2x^2 + -2x + -2 = 0 | 1. Identify coefficients: a=2, b=-2, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -16 = 20.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √20) / 4.
5. Calculate roots: x1 = 1.618033988749895, x2 = -0.6180339887498949.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[2, -5]
[-5, 5]
B =
[-5, -1]
[5, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-5) + (-5*5) = -35
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (-5*1) = -7
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (5*5) = 50
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (5*1) = 10
Resu... |
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) |
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) |
Compute the quotient and remainder of 96205 divided by 95. Show the long division steps. | To divide 96205 by 95:
1. Determine how many times 95 goes into the leading digits of 96205.
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: 96205 / 95 = 1012 with a remainder of 65.
Proof: (1012 * 95... |
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) = (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 derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 18874 divided by 77. Show the long division steps. | To divide 18874 by 77:
1. Determine how many times 77 goes into the leading digits of 18874.
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: 18874 / 77 = 245 with a remainder of 9.
Proof: (245 * 77) +... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 256 + 25 = c^2
3. Sum the squares: 281 = c^2
4. Take the square root of both sides: c = √281
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 9, 6, 16, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 9 + 6 + 16 + 19) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.4, -1.4, -4.4, 5.6, 8.6]
Squared Differences: [70.56, 1.96, 19.36, 31.36, 73.96]
... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 8^2 = c^2
2. Calculate the squares: 9 + 64 = c^2
3. Sum the squares: 73 = c^2
4. Take the square root of both sides: c = √73
Final Answer: ... |
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 derivative of f(x) = (3x + 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 = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
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) |
Multiply the following 2x2 matrices:
A =
[3, 0]
[-5, -1]
B =
[3, 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*3) + (0*-5) = 9
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (0*5) = 12
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (-1*-5) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (-1*5) = -25
Result... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 4. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 4^2 = c^2
2. Calculate the squares: 256 + 16 = c^2
3. Sum the squares: 272 = c^2
4. Take the square root of both sides: c = √272
Final Ans... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 8, 10, 12, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 8 + 10 + 12 + 3) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, -1.4, 0.6, 2.6, -6.4]
Squared Differences: [21.16, 1.96, 0.36, 6.76, 40.96]
... |
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 |
Compute the quotient and remainder of 89556 divided by 9. Show the long division steps. | To divide 89556 by 9:
1. Determine how many times 9 goes into the leading digits of 89556.
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: 89556 / 9 = 9950 with a remainder of 6.
Proof: (9950 * 9) + 6... |
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 ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 3, 17, 19, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 3 + 17 + 19 + 17) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, -10.0, 4.0, 6.0, 4.0]
Squared Differences: [16.0, 100.0, 16.0, 36.0, 16.0]
... |
Add the fractions 6/11 and 6/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 6, which is 66.
2. Convert 6/11 by multiplying top and bottom by 6: 36/66.
3. Convert 6/6 by multiplying top and bottom by 11: 66/66.
4. Add the numerators: 36 + 66 = 102.
5. The resulting fraction is 102/66.
6. Simplify b... |
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 34824 divided by 82. Show the long division steps. | To divide 34824 by 82:
1. Determine how many times 82 goes into the leading digits of 34824.
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: 34824 / 82 = 424 with a remainder of 56.
Proof: (424 * 82) ... |
Add the fractions 11/6 and 5/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 9, which is 18.
2. Convert 11/6 by multiplying top and bottom by 3: 33/18.
3. Convert 5/9 by multiplying top and bottom by 2: 10/18.
4. Add the numerators: 33 + 10 = 43.
5. The resulting fraction is 43/18.
Final Answer: 43/... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 14^2 = c^2
2. Calculate the squares: 100 + 196 = c^2
3. Sum the squares: 296 = c^2
4. Take the square root of both sides: c = √296
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 16^2 = c^2
2. Calculate the squares: 100 + 256 = c^2
3. Sum the squares: 356 = c^2
4. Take the square root of both sides: c = √356
Final A... |
Compute the quotient and remainder of 16698 divided by 88. Show the long division steps. | To divide 16698 by 88:
1. Determine how many times 88 goes into the leading digits of 16698.
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: 16698 / 88 = 189 with a remainder of 66.
Proof: (189 * 88) ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 13^2 = c^2
2. Calculate the squares: 400 + 169 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Compute the quotient and remainder of 60291 divided by 16. Show the long division steps. | To divide 60291 by 16:
1. Determine how many times 16 goes into the leading digits of 60291.
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: 60291 / 16 = 3768 with a remainder of 3.
Proof: (3768 * 16)... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 20, 6, 20, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 20 + 6 + 20 + 12) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, 6.6, -7.4, 6.6, -1.4]
Squared Differences: [19.36, 43.56, 54.76, 43.56, 1.96]... |
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) |
Compute the quotient and remainder of 12985 divided by 14. Show the long division steps. | To divide 12985 by 14:
1. Determine how many times 14 goes into the leading digits of 12985.
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: 12985 / 14 = 927 with a remainder of 7.
Proof: (927 * 14) +... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 16^2 = c^2
2. Calculate the squares: 36 + 256 = c^2
3. Sum the squares: 292 = c^2
4. Take the square root of both sides: c = √292
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 13^2 = c^2
2. Calculate the squares: 196 + 169 = c^2
3. Sum the squares: 365 = c^2
4. Take the square root of both sides: c = √365
Final A... |
Find the roots of the quadratic equation: 7x^2 + 2x + 10 = 0 | 1. Identify coefficients: a=7, b=2, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(7)(10) = 4 - 280 = -276.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √276i) / 14.
Final Answer: x = -0.14285714285714285 ± 1.1866605518454392i |
Find the roots of the quadratic equation: 4x^2 + -4x + -8 = 0 | 1. Identify coefficients: a=4, b=-4, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -128 = 144.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √144) / 8.
5. Calculate roots: x1 = 2.0, x2 = -1.0.
Final Answer: x = 2.0, -1.0 |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 12, 3, 17, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 12 + 3 + 17 + 10) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, 0.6, -8.4, 5.6, -1.4]
Squared Differences: [12.96, 0.36, 70.56, 31.36, 1.96]
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.