prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Multiply the following 2x2 matrices:
A =
[3, 2]
[-1, -1]
B =
[2, 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) = (3*2) + (2*5) = 16
C12 = (A11 * B12) + (A12 * B22) = (3*5) + (2*1) = 17
C21 = (A21 * B11) + (A22 * B21) = (-1*2) + (-1*5) = -7
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (-1*1) = -6
Result Ma... |
Find the roots of the quadratic equation: 1x^2 + -6x + 2 = 0 | 1. Identify coefficients: a=1, b=-6, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 8 = 28.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √28) / 2.
5. Calculate roots: x1 = 5.645751311064591, x2 = 0.3542486889354093.
Final Answer: ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 12^2 = c^2
2. Calculate the squares: 169 + 144 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
Find the roots of the quadratic equation: 10x^2 + 4x + 10 = 0 | 1. Identify coefficients: a=10, b=4, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(10)(10) = 16 - 400 = -384.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √384i) / 20.
Final Answer: x = -0.2 ± 0.9797958971132712i |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 18, 1, 9, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 18 + 1 + 9 + 6) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.0, 9.0, -8.0, 0.0, -3.0]
Squared Differences: [4.0, 81.0, 64.0, 0.0, 9.0]
Sum of... |
Compute the quotient and remainder of 82600 divided by 19. Show the long division steps. | To divide 82600 by 19:
1. Determine how many times 19 goes into the leading digits of 82600.
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: 82600 / 19 = 4347 with a remainder of 7.
Proof: (4347 * 19)... |
Compute the quotient and remainder of 40568 divided by 77. Show the long division steps. | To divide 40568 by 77:
1. Determine how many times 77 goes into the leading digits of 40568.
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: 40568 / 77 = 526 with a remainder of 66.
Proof: (526 * 77) ... |
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 |
Compute the quotient and remainder of 71851 divided by 72. Show the long division steps. | To divide 71851 by 72:
1. Determine how many times 72 goes into the leading digits of 71851.
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: 71851 / 72 = 997 with a remainder of 67.
Proof: (997 * 72) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 15, 10, 12, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 15 + 10 + 12 + 17) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, 1.4, -3.6, -1.6, 3.4]
Squared Differences: [0.16, 1.96, 12.96, 2.56, 11.56]
... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 6^2 + 17^2 = c^2
2. Calculate the squares: 36 + 289 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
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 |
Add the fractions 11/13 and 12/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 2, which is 26.
2. Convert 11/13 by multiplying top and bottom by 2: 22/26.
3. Convert 12/2 by multiplying top and bottom by 13: 156/26.
4. Add the numerators: 22 + 156 = 178.
5. The resulting fraction is 178/26.
6. Simpli... |
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) = 6x^2. | To find the indefinite integral ∫ 6x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + C |
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 indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^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: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[4, 5]
B =
[3, 4]
[-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) = (-5*3) + (0*-3) = -15
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (0*4) = -20
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (5*-3) = -3
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (5*4) = 36
Result ... |
Add the fractions 6/10 and 13/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 10, which is 10.
2. Convert 6/10 by multiplying top and bottom by 1: 6/10.
3. Convert 13/10 by multiplying top and bottom by 1: 13/10.
4. Add the numerators: 6 + 13 = 19.
5. The resulting fraction is 19/10.
Final Answer: 1... |
Compute the quotient and remainder of 87067 divided by 51. Show the long division steps. | To divide 87067 by 51:
1. Determine how many times 51 goes into the leading digits of 87067.
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: 87067 / 51 = 1707 with a remainder of 10.
Proof: (1707 * 51... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 11^2 = c^2
2. Calculate the squares: 64 + 121 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 14, 9, 19, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 14 + 9 + 19 + 4) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.4, 4.6, -0.4, 9.6, -5.4]
Squared Differences: [70.56, 21.16, 0.16, 92.16, 29.16]
... |
Compute the quotient and remainder of 20291 divided by 61. Show the long division steps. | To divide 20291 by 61:
1. Determine how many times 61 goes into the leading digits of 20291.
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: 20291 / 61 = 332 with a remainder of 39.
Proof: (332 * 61) ... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 11^2 = c^2
2. Calculate the squares: 100 + 121 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 10^2 = c^2
2. Calculate the squares: 25 + 100 = c^2
3. Sum the squares: 125 = c^2
4. Take the square root of both sides: c = √125
Final Ans... |
Compute the quotient and remainder of 18441 divided by 13. Show the long division steps. | To divide 18441 by 13:
1. Determine how many times 13 goes into the leading digits of 18441.
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: 18441 / 13 = 1418 with a remainder of 7.
Proof: (1418 * 13)... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 10, 15, 13, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 10 + 15 + 13 + 13) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -2.2, 2.8, 0.8, 0.8]
Squared Differences: [4.84, 4.84, 7.84, 0.64, 0.64]
... |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 4^2 + 11^2 = c^2
2. Calculate the squares: 16 + 121 = c^2
3. Sum the squares: 137 = c^2
4. Take the square root of both sides: c = √137
Final Ans... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[-5, -3]
B =
[5, 0]
[-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) = (-1*5) + (-4*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (-4*3) = -12
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (-3*-1) = -22
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (-3*3) = -9
R... |
Add the fractions 15/11 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 9, which is 99.
2. Convert 15/11 by multiplying top and bottom by 9: 135/99.
3. Convert 15/9 by multiplying top and bottom by 11: 165/99.
4. Add the numerators: 135 + 165 = 300.
5. The resulting fraction is 300/99.
6. Simp... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 10, 11, 7, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 10 + 11 + 7 + 2) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, 2.4, 3.4, -0.6, -5.6]
Squared Differences: [0.16, 5.76, 11.56, 0.36, 31.36]
S... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 9^2 = c^2
2. Calculate the squares: 169 + 81 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
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) |
Compute the quotient and remainder of 88590 divided by 24. Show the long division steps. | To divide 88590 by 24:
1. Determine how many times 24 goes into the leading digits of 88590.
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: 88590 / 24 = 3691 with a remainder of 6.
Proof: (3691 * 24)... |
Multiply the following 2x2 matrices:
A =
[3, -1]
[-2, -4]
B =
[-4, 5]
[1, 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*-4) + (-1*1) = -13
C12 = (A11 * B12) + (A12 * B22) = (3*5) + (-1*2) = 13
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (-4*1) = 4
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (-4*2) = -18
Resu... |
Find the roots of the quadratic equation: 8x^2 + 4x + 0 = 0 | 1. Identify coefficients: a=8, b=4, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - 0 = 16.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √16) / 16.
5. Calculate roots: x1 = 0.0, x2 = -0.5.
Final Answer: x = 0.0, -0.5 |
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 indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^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: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
Find the roots of the quadratic equation: 4x^2 + -9x + -9 = 0 | 1. Identify coefficients: a=4, b=-9, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -144 = 225.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √225) / 8.
5. Calculate roots: x1 = 3.0, x2 = -0.75.
Final Answer: x = 3.0, -0.75 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 1, 13, 11, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 1 + 13 + 11 + 6) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, -5.8, 6.2, 4.2, -0.8]
Squared Differences: [14.44, 33.64, 38.44, 17.64, 0.64]
... |
Add the fractions 6/12 and 3/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 7, which is 84.
2. Convert 6/12 by multiplying top and bottom by 7: 42/84.
3. Convert 3/7 by multiplying top and bottom by 12: 36/84.
4. Add the numerators: 42 + 36 = 78.
5. The resulting fraction is 78/84.
6. Simplify by ... |
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 ... |
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) |
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 +... |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[5, 0]
B =
[5, -4]
[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) = (-1*5) + (1*5) = 0
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (1*3) = 7
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (0*5) = 25
C22 = (A21 * B12) + (A22 * B22) = (5*-4) + (0*3) = -20
Result Mat... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 17, 2, 10, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 17 + 2 + 10 + 16) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.0, 7.0, -8.0, 0.0, 6.0]
Squared Differences: [25.0, 49.0, 64.0, 0.0, 36.0]
Su... |
Add the fractions 9/11 and 12/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 11, which is 11.
2. Convert 9/11 by multiplying top and bottom by 1: 9/11.
3. Convert 12/11 by multiplying top and bottom by 1: 12/11.
4. Add the numerators: 9 + 12 = 21.
5. The resulting fraction is 21/11.
Final Answer: 2... |
Find the roots of the quadratic equation: 5x^2 + -1x + 4 = 0 | 1. Identify coefficients: a=5, b=-1, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(5)(4) = 1 - 80 = -79.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √79i) / 10.
Final Answer: x = 0.1 ± 0.8888194417315589i |
Add the fractions 14/12 and 12/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 4, which is 12.
2. Convert 14/12 by multiplying top and bottom by 1: 14/12.
3. Convert 12/4 by multiplying top and bottom by 3: 36/12.
4. Add the numerators: 14 + 36 = 50.
5. The resulting fraction is 50/12.
6. Simplify by... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 5, 2, 19, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 5 + 2 + 19 + 13) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.6, -6.4, -9.4, 7.6, 1.6]
Squared Differences: [43.56, 40.96, 88.36, 57.76, 2.56]
... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + -4x + -10 = 0 | 1. Identify coefficients: a=8, b=-4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -320 = 336.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √336) / 16.
5. Calculate roots: x1 = 1.39564392373896, x2 = -0.89564392373896.
Final An... |
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 =
[-2, 1]
[3, -1]
B =
[3, 3]
[0, 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*3) + (1*0) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (1*2) = -4
C21 = (A21 * B11) + (A22 * B21) = (3*3) + (-1*0) = 9
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (-1*2) = 7
Result Matr... |
Compute the quotient and remainder of 26554 divided by 79. Show the long division steps. | To divide 26554 by 79:
1. Determine how many times 79 goes into the leading digits of 26554.
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: 26554 / 79 = 336 with a remainder of 10.
Proof: (336 * 79) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 1, 17, 19, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 1 + 17 + 19 + 14) / 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, -12.4, 3.6, 5.6, 0.6]
Squared Differences: [6.76, 153.76, 12.96, 31.36, 0.36]... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[-4, -1]
B =
[1, -3]
[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) = (-3*1) + (5*3) = 12
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (5*3) = 24
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (-1*3) = -7
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (-1*3) = 9
Result... |
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 |
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 |
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) |
Find the roots of the quadratic equation: 7x^2 + -4x + 8 = 0 | 1. Identify coefficients: a=7, b=-4, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(7)(8) = 16 - 224 = -208.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √208i) / 14.
Final Answer: x = 0.2857142857142857 ± 1.0301575072754254i |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Compute the quotient and remainder of 52982 divided by 42. Show the long division steps. | To divide 52982 by 42:
1. Determine how many times 42 goes into the leading digits of 52982.
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: 52982 / 42 = 1261 with a remainder of 20.
Proof: (1261 * 42... |
Add the fractions 10/2 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 13, which is 26.
2. Convert 10/2 by multiplying top and bottom by 13: 130/26.
3. Convert 5/13 by multiplying top and bottom by 2: 10/26.
4. Add the numerators: 130 + 10 = 140.
5. The resulting fraction is 140/26.
6. Simplif... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 12, 3, 8, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 12 + 3 + 8 + 9) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.6, 1.6, -7.4, -2.4, -1.4]
Squared Differences: [92.16, 2.56, 54.76, 5.76, 1.96]
... |
Find the roots of the quadratic equation: 8x^2 + 10x + -10 = 0 | 1. Identify coefficients: a=8, b=10, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -320 = 420.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √420) / 16.
5. Calculate roots: x1 = 0.6558688457449497, x2 = -1.9058688457449497.
F... |
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 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 |
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 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 |
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 ... |
Add the fractions 12/10 and 11/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 6, which is 30.
2. Convert 12/10 by multiplying top and bottom by 3: 36/30.
3. Convert 11/6 by multiplying top and bottom by 5: 55/30.
4. Add the numerators: 36 + 55 = 91.
5. The resulting fraction is 91/30.
Final Answer: ... |
Find the roots of the quadratic equation: 7x^2 + -6x + 3 = 0 | 1. Identify coefficients: a=7, b=-6, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(7)(3) = 36 - 84 = -48.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √48i) / 14.
Final Answer: x = 0.42857142857142855 ± 0.4948716593053935i |
Compute the quotient and remainder of 15641 divided by 18. Show the long division steps. | To divide 15641 by 18:
1. Determine how many times 18 goes into the leading digits of 15641.
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: 15641 / 18 = 868 with a remainder of 17.
Proof: (868 * 18) ... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (2x + 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 = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Add the fractions 8/2 and 2/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 9, which is 18.
2. Convert 8/2 by multiplying top and bottom by 9: 72/18.
3. Convert 2/9 by multiplying top and bottom by 2: 4/18.
4. Add the numerators: 72 + 4 = 76.
5. The resulting fraction is 76/18.
6. Simplify by divid... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 8, 4, 1, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 8 + 4 + 1 + 11) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, 1.2, -2.8, -5.8, 4.2]
Squared Differences: [10.24, 1.44, 7.84, 33.64, 17.64]
... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[-3, 5]
B =
[1, 0]
[-4, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*1) + (5*-4) = -21
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (5*-5) = -25
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (5*-4) = -23
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (5*-5) = -25
R... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 14^2 = c^2
2. Calculate the squares: 324 + 196 = c^2
3. Sum the squares: 520 = c^2
4. Take the square root of both sides: c = √520
Final A... |
Add the fractions 11/6 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 6, which is 6.
2. Convert 11/6 by multiplying top and bottom by 1: 11/6.
3. Convert 3/6 by multiplying top and bottom by 1: 3/6.
4. Add the numerators: 11 + 3 = 14.
5. The resulting fraction is 14/6.
6. Simplify by dividing... |
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 +... |
Add the fractions 13/11 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 3, which is 33.
2. Convert 13/11 by multiplying top and bottom by 3: 39/33.
3. Convert 14/3 by multiplying top and bottom by 11: 154/33.
4. Add the numerators: 39 + 154 = 193.
5. The resulting fraction is 193/33.
Final Ans... |
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 +... |
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 88761 divided by 18. Show the long division steps. | To divide 88761 by 18:
1. Determine how many times 18 goes into the leading digits of 88761.
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: 88761 / 18 = 4931 with a remainder of 3.
Proof: (4931 * 18)... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 5^2 = c^2
2. Calculate the squares: 16 + 25 = c^2
3. Sum the squares: 41 = c^2
4. Take the square root of both sides: c = √41
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[0, 4]
[2, -1]
B =
[-5, 3]
[5, 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) = (0*-5) + (4*5) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*3) + (4*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-1*5) = -15
C22 = (A21 * B12) + (A22 * B22) = (2*3) + (-1*0) = 6
Result Mat... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[5, 4]
B =
[-4, -5]
[4, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-4) + (1*4) = 16
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (1*1) = 16
C21 = (A21 * B11) + (A22 * B21) = (5*-4) + (4*4) = -4
C22 = (A21 * B12) + (A22 * B22) = (5*-5) + (4*1) = -21
Result... |
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: 2x^2 + -10x + 3 = 0 | 1. Identify coefficients: a=2, b=-10, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 24 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √76) / 4.
5. Calculate roots: x1 = 4.6794494717703365, x2 = 0.320550528229663.
Final Answ... |
Add the fractions 14/8 and 13/14. 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 14/8 by multiplying top and bottom by 7: 98/56.
3. Convert 13/14 by multiplying top and bottom by 4: 52/56.
4. Add the numerators: 98 + 52 = 150.
5. The resulting fraction is 150/56.
6. Simplify ... |
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) |
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 indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + C |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.