prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
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 ... |
Multiply the following 2x2 matrices:
A =
[3, -2]
[-4, 2]
B =
[-2, 1]
[-3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*-2) + (-2*-3) = 0
C12 = (A11 * B12) + (A12 * B22) = (3*1) + (-2*-2) = 7
C21 = (A21 * B11) + (A22 * B21) = (-4*-2) + (2*-3) = 2
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (2*-2) = -8
Result... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 10^2 = c^2
2. Calculate the squares: 289 + 100 = c^2
3. Sum the squares: 389 = c^2
4. Take the square root of both sides: c = √389
Final A... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[-3, -2]
B =
[-4, 2]
[-3, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-4) + (2*-3) = -26
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (2*1) = 12
C21 = (A21 * B11) + (A22 * B21) = (-3*-4) + (-2*-3) = 18
C22 = (A21 * B12) + (A22 * B22) = (-3*2) + (-2*1) = -8
Resu... |
Multiply the following 2x2 matrices:
A =
[0, -5]
[4, -4]
B =
[4, 2]
[-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) = (0*4) + (-5*-4) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (-5*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*4) + (-4*-4) = 32
C22 = (A21 * B12) + (A22 * B22) = (4*2) + (-4*0) = 8
Result Ma... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
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 |
Multiply the following 2x2 matrices:
A =
[4, 5]
[-5, -3]
B =
[4, -4]
[3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*4) + (5*3) = 31
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (5*-2) = -26
C21 = (A21 * B11) + (A22 * B21) = (-5*4) + (-3*3) = -29
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (-3*-2) = 26
Res... |
Find the roots of the quadratic equation: 6x^2 + -7x + 6 = 0 | 1. Identify coefficients: a=6, b=-7, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(6)(6) = 49 - 144 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √95i) / 12.
Final Answer: x = 0.5833333333333334 ± 0.8122328620674136i |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 10, 20, 11, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 10 + 20 + 11 + 5) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -2.6, 7.4, -1.6, -7.6]
Squared Differences: [19.36, 6.76, 54.76, 2.56, 57.76]... |
Find the roots of the quadratic equation: 3x^2 + 2x + -6 = 0 | 1. Identify coefficients: a=3, b=2, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -72 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √76) / 6.
5. Calculate roots: x1 = 1.1196329811802246, x2 = -1.7862996478468913.
Final Answ... |
Compute the quotient and remainder of 80074 divided by 85. Show the long division steps. | To divide 80074 by 85:
1. Determine how many times 85 goes into the leading digits of 80074.
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: 80074 / 85 = 942 with a remainder of 4.
Proof: (942 * 85) +... |
Find the roots of the quadratic equation: 5x^2 + 7x + -9 = 0 | 1. Identify coefficients: a=5, b=7, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -180 = 229.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √229) / 10.
5. Calculate roots: x1 = 0.8132745950421556, x2 = -2.2132745950421557.
Final... |
Find the roots of the quadratic equation: 5x^2 + 0x + -4 = 0 | 1. Identify coefficients: a=5, b=0, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -80 = 80.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √80) / 10.
5. Calculate roots: x1 = 0.894427190999916, x2 = -0.894427190999916.
Final Answer... |
Find the roots of the quadratic equation: 5x^2 + -6x + 2 = 0 | 1. Identify coefficients: a=5, b=-6, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(5)(2) = 36 - 40 = -4.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √4i) / 10.
Final Answer: x = 0.6 ± 0.2i |
Compute the quotient and remainder of 45890 divided by 77. Show the long division steps. | To divide 45890 by 77:
1. Determine how many times 77 goes into the leading digits of 45890.
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: 45890 / 77 = 595 with a remainder of 75.
Proof: (595 * 77) ... |
Compute the quotient and remainder of 78841 divided by 45. Show the long division steps. | To divide 78841 by 45:
1. Determine how many times 45 goes into the leading digits of 78841.
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: 78841 / 45 = 1752 with a remainder of 1.
Proof: (1752 * 45)... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 7^2 = c^2
2. Calculate the squares: 361 + 49 = c^2
3. Sum the squares: 410 = c^2
4. Take the square root of both sides: c = √410
Final Ans... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Add the fractions 14/2 and 14/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 5, which is 10.
2. Convert 14/2 by multiplying top and bottom by 5: 70/10.
3. Convert 14/5 by multiplying top and bottom by 2: 28/10.
4. Add the numerators: 70 + 28 = 98.
5. The resulting fraction is 98/10.
6. Simplify by d... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 13, 19, 20, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 13 + 19 + 20 + 20) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-13.6, -1.6, 4.4, 5.4, 5.4]
Squared Differences: [184.96, 2.56, 19.36, 29.16, 29.1... |
Add the fractions 9/15 and 9/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 11, which is 165.
2. Convert 9/15 by multiplying top and bottom by 11: 99/165.
3. Convert 9/11 by multiplying top and bottom by 15: 135/165.
4. Add the numerators: 99 + 135 = 234.
5. The resulting fraction is 234/165.
6. S... |
Find the roots of the quadratic equation: 10x^2 + 4x + 9 = 0 | 1. Identify coefficients: a=10, b=4, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(10)(9) = 16 - 360 = -344.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √344i) / 20.
Final Answer: x = -0.2 ± 0.9273618495495703i |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 13^2 = c^2
2. Calculate the squares: 144 + 169 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 13^2 = c^2
2. Calculate the squares: 256 + 169 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final A... |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^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: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[2, -5]
[0, 0]
B =
[-1, -5]
[-4, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-1) + (-5*-4) = 18
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-5*2) = -20
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (0*-4) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (0*2) = 0
Result... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^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: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^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 +... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[-3, -2]
B =
[-2, -2]
[-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) = (-1*-2) + (5*-5) = -23
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (5*-5) = -23
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (-2*-5) = 16
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (-2*-5) = 1... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 2, 7, 5, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 2 + 7 + 5 + 5) / 5 = 24 / 5 = 4.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, -2.8, 2.2, 0.2, 0.2]
Squared Differences: [0.04, 7.84, 4.84, 0.04, 0.04]
Sum of... |
Find the roots of the quadratic equation: 9x^2 + -6x + 5 = 0 | 1. Identify coefficients: a=9, b=-6, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(9)(5) = 36 - 180 = -144.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √144i) / 18.
Final Answer: x = 0.3333333333333333 ± 0.6666666666666666i |
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 |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Multiply the following 2x2 matrices:
A =
[-5, 5]
[2, 4]
B =
[5, -3]
[-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) = (-5*5) + (5*-3) = -40
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (5*5) = 40
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (4*-3) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*-3) + (4*5) = 14
Result... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 9^2 = c^2
2. Calculate the squares: 81 + 81 = c^2
3. Sum the squares: 162 = c^2
4. Take the square root of both sides: c = √162
Final Answe... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 10, 12, 4, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 10 + 12 + 4 + 1) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.6, 0.6, 2.6, -5.4, -8.4]
Squared Differences: [112.36, 0.36, 6.76, 29.16, 70.56]
... |
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 |
Add the fractions 8/9 and 13/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 7, which is 63.
2. Convert 8/9 by multiplying top and bottom by 7: 56/63.
3. Convert 13/7 by multiplying top and bottom by 9: 117/63.
4. Add the numerators: 56 + 117 = 173.
5. The resulting fraction is 173/63.
Final Answer:... |
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 +... |
Find the roots of the quadratic equation: 6x^2 + 3x + 6 = 0 | 1. Identify coefficients: a=6, b=3, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(6)(6) = 9 - 144 = -135.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √135i) / 12.
Final Answer: x = -0.25 ± 0.9682458365518541i |
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) |
Add the fractions 9/15 and 9/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 10, which is 30.
2. Convert 9/15 by multiplying top and bottom by 2: 18/30.
3. Convert 9/10 by multiplying top and bottom by 3: 27/30.
4. Add the numerators: 18 + 27 = 45.
5. The resulting fraction is 45/30.
6. Simplify by... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 7, 15, 19, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 7 + 15 + 19 + 9) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -6.6, 1.4, 5.4, -4.6]
Squared Differences: [19.36, 43.56, 1.96, 29.16, 21.16]
... |
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 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 16^2 = c^2
2. Calculate the squares: 324 + 256 = c^2
3. Sum the squares: 580 = c^2
4. Take the square root of both sides: c = √580
Final A... |
Find the roots of the quadratic equation: 3x^2 + 8x + -6 = 0 | 1. Identify coefficients: a=3, b=8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -72 = 136.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √136) / 6.
5. Calculate roots: x1 = 0.610317298281767, x2 = -3.2769839649484336.
Final An... |
Find the roots of the quadratic equation: 6x^2 + -9x + -5 = 0 | 1. Identify coefficients: a=6, b=-9, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -120 = 201.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √201) / 12.
5. Calculate roots: x1 = 1.9314539065631522, x2 = -0.43145390656315213.
Fina... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 10, 7, 11, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 10 + 7 + 11 + 17) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, 0.2, -2.8, 1.2, 7.2]
Squared Differences: [33.64, 0.04, 7.84, 1.44, 51.84]
... |
Multiply the following 2x2 matrices:
A =
[3, -5]
[-5, -4]
B =
[1, -3]
[-4, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*1) + (-5*-4) = 23
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (-5*-4) = 11
C21 = (A21 * B11) + (A22 * B21) = (-5*1) + (-4*-4) = 11
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (-4*-4) = 31
R... |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[3, -4]
[-5, 1]
B =
[-4, -4]
[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) = (3*-4) + (-4*3) = -24
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (-4*-5) = 8
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (1*3) = 23
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (1*-5) = 15
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 1, 15, 11, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 1 + 15 + 11 + 18) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, -11.2, 2.8, -1.2, 5.8]
Squared Differences: [14.44, 125.44, 7.84, 1.44, 33.64... |
Compute the quotient and remainder of 66739 divided by 22. Show the long division steps. | To divide 66739 by 22:
1. Determine how many times 22 goes into the leading digits of 66739.
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: 66739 / 22 = 3033 with a remainder of 13.
Proof: (3033 * 22... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 16^2 = c^2
2. Calculate the squares: 25 + 256 = c^2
3. Sum the squares: 281 = c^2
4. Take the square root of both sides: c = √281
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 13^2 = c^2
2. Calculate the squares: 25 + 169 = c^2
3. Sum the squares: 194 = c^2
4. Take the square root of both sides: c = √194
Final Ans... |
Compute the quotient and remainder of 79483 divided by 56. Show the long division steps. | To divide 79483 by 56:
1. Determine how many times 56 goes into the leading digits of 79483.
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: 79483 / 56 = 1419 with a remainder of 19.
Proof: (1419 * 56... |
Add the fractions 4/13 and 9/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 6, which is 78.
2. Convert 4/13 by multiplying top and bottom by 6: 24/78.
3. Convert 9/6 by multiplying top and bottom by 13: 117/78.
4. Add the numerators: 24 + 117 = 141.
5. The resulting fraction is 141/78.
6. Simplify... |
Find the roots of the quadratic equation: 5x^2 + -9x + 6 = 0 | 1. Identify coefficients: a=5, b=-9, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(5)(6) = 81 - 120 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √39i) / 10.
Final Answer: x = 0.9 ± 0.6244997998398398i |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 16^2 = c^2
2. Calculate the squares: 196 + 256 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[2, -3]
B =
[2, 5]
[-3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*2) + (-4*-3) = 12
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (-4*-2) = 8
C21 = (A21 * B11) + (A22 * B21) = (2*2) + (-3*-3) = 13
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (-3*-2) = 16
Result... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 66884 divided by 14. Show the long division steps. | To divide 66884 by 14:
1. Determine how many times 14 goes into the leading digits of 66884.
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: 66884 / 14 = 4777 with a remainder of 6.
Proof: (4777 * 14)... |
Compute the quotient and remainder of 10314 divided by 99. Show the long division steps. | To divide 10314 by 99:
1. Determine how many times 99 goes into the leading digits of 10314.
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: 10314 / 99 = 104 with a remainder of 18.
Proof: (104 * 99) ... |
Multiply the following 2x2 matrices:
A =
[0, 4]
[3, -4]
B =
[-4, -2]
[-4, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*-4) + (4*-4) = -16
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (4*3) = 12
C21 = (A21 * B11) + (A22 * B21) = (3*-4) + (-4*-4) = 4
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-4*3) = -18
Resu... |
Add the fractions 8/6 and 12/5. 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 12/5 by multiplying top and bottom by 6: 72/30.
4. Add the numerators: 40 + 72 = 112.
5. The resulting fraction is 112/30.
6. Simplify by ... |
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: [9, 13, 8, 13, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 13 + 8 + 13 + 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, -4.0, 1.0, 5.0]
Squared Differences: [9.0, 1.0, 16.0, 1.0, 25.0]
Sum ... |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^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: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Add the fractions 3/6 and 9/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 3, which is 6.
2. Convert 3/6 by multiplying top and bottom by 1: 3/6.
3. Convert 9/3 by multiplying top and bottom by 2: 18/6.
4. Add the numerators: 3 + 18 = 21.
5. The resulting fraction is 21/6.
6. Simplify by dividing ... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 14^2 = c^2
2. Calculate the squares: 225 + 196 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 5, 4, 18, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 5 + 4 + 18 + 1) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -1.2, -2.2, 11.8, -5.2]
Squared Differences: [10.24, 1.44, 4.84, 139.24, 27.04]
... |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 6^2 + 19^2 = c^2
2. Calculate the squares: 36 + 361 = c^2
3. Sum the squares: 397 = c^2
4. Take the square root of both sides: c = √397
Final Ans... |
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 62842 divided by 41. Show the long division steps. | To divide 62842 by 41:
1. Determine how many times 41 goes into the leading digits of 62842.
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: 62842 / 41 = 1532 with a remainder of 30.
Proof: (1532 * 41... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 3, 19, 2, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 3 + 19 + 2 + 1) / 5 = 27 / 5 = 5.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, -2.4, 13.6, -3.4, -4.4]
Squared Differences: [11.56, 5.76, 184.96, 11.56, 19.36]... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 7x^2 + 2x + 7 = 0 | 1. Identify coefficients: a=7, b=2, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(7)(7) = 4 - 196 = -192.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √192i) / 14.
Final Answer: x = -0.14285714285714285 ± 0.989743318610787i |
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 ... |
Multiply the following 2x2 matrices:
A =
[4, -5]
[-3, 1]
B =
[-3, 3]
[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) = (4*-3) + (-5*5) = -37
C12 = (A11 * B12) + (A12 * B22) = (4*3) + (-5*3) = -3
C21 = (A21 * B11) + (A22 * B21) = (-3*-3) + (1*5) = 14
C22 = (A21 * B12) + (A22 * B22) = (-3*3) + (1*3) = -6
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 20, 20, 20, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 20 + 20 + 20 + 11) / 5 = 85 / 5 = 17.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, 3.0, 3.0, 3.0, -6.0]
Squared Differences: [9.0, 9.0, 9.0, 9.0, 36.0]
Sum... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[1, 2]
[1, 1]
B =
[5, 4]
[-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) = (1*5) + (2*-4) = -3
C12 = (A11 * B12) + (A12 * B22) = (1*4) + (2*0) = 4
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (1*-4) = 1
C22 = (A21 * B12) + (A22 * B22) = (1*4) + (1*0) = 4
Result Matrix ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 18^2 = c^2
2. Calculate the squares: 400 + 324 = c^2
3. Sum the squares: 724 = c^2
4. Take the square root of both sides: c = √724
Final A... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 17^2 = c^2
2. Calculate the squares: 361 + 289 = c^2
3. Sum the squares: 650 = c^2
4. Take the square root of both sides: c = √650
Final A... |
Compute the quotient and remainder of 5969 divided by 79. Show the long division steps. | To divide 5969 by 79:
1. Determine how many times 79 goes into the leading digits of 5969.
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: 5969 / 79 = 75 with a remainder of 44.
Proof: (75 * 79) + 44 ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 15^2 = c^2
2. Calculate the squares: 169 + 225 = c^2
3. Sum the squares: 394 = c^2
4. Take the square root of both sides: c = √394
Final A... |
Compute the quotient and remainder of 41454 divided by 64. Show the long division steps. | To divide 41454 by 64:
1. Determine how many times 64 goes into the leading digits of 41454.
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: 41454 / 64 = 647 with a remainder of 46.
Proof: (647 * 64) ... |
Find the roots of the quadratic equation: 1x^2 + -9x + 9 = 0 | 1. Identify coefficients: a=1, b=-9, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 36 = 45.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √45) / 2.
5. Calculate roots: x1 = 7.854101966249685, x2 = 1.1458980337503153.
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[-3, -5]
B =
[-4, -1]
[5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-4) + (2*5) = -10
C12 = (A11 * B12) + (A12 * B22) = (5*-1) + (2*-2) = -9
C21 = (A21 * B11) + (A22 * B21) = (-3*-4) + (-5*5) = -13
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (-5*-2) = 13
R... |
Compute the indefinite integral of f(x) = 2x^1. | To find the indefinite integral ∫ 2x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^2 + C |
Add the fractions 9/12 and 4/4. 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 9/12 by multiplying top and bottom by 1: 9/12.
3. Convert 4/4 by multiplying top and bottom by 3: 12/12.
4. Add the numerators: 9 + 12 = 21.
5. The resulting fraction is 21/12.
6. Simplify by div... |
Find the roots of the quadratic equation: 5x^2 + 10x + -5 = 0 | 1. Identify coefficients: a=5, b=10, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -100 = 200.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √200) / 10.
5. Calculate roots: x1 = 0.4142135623730951, x2 = -2.414213562373095.
Fin... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.