prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Add the fractions 7/13 and 9/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 15, which is 195.
2. Convert 7/13 by multiplying top and bottom by 15: 105/195.
3. Convert 9/15 by multiplying top and bottom by 13: 117/195.
4. Add the numerators: 105 + 117 = 222.
5. The resulting fraction is 222/195.
6.... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 5^2 = c^2
2. Calculate the squares: 64 + 25 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
Find the derivative of f(x) = (3x + 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 = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
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 =
[4, 5]
[-3, 4]
B =
[0, 5]
[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) = (4*0) + (5*3) = 15
C12 = (A11 * B12) + (A12 * B22) = (4*5) + (5*1) = 25
C21 = (A21 * B11) + (A22 * B21) = (-3*0) + (4*3) = 12
C22 = (A21 * B12) + (A22 * B22) = (-3*5) + (4*1) = -11
Result Mat... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Evaluate tan(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) = (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 +... |
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 78439 divided by 9. Show the long division steps. | To divide 78439 by 9:
1. Determine how many times 9 goes into the leading digits of 78439.
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: 78439 / 9 = 8715 with a remainder of 4.
Proof: (8715 * 9) + 4... |
Add the fractions 2/8 and 3/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 11, which is 88.
2. Convert 2/8 by multiplying top and bottom by 11: 22/88.
3. Convert 3/11 by multiplying top and bottom by 8: 24/88.
4. Add the numerators: 22 + 24 = 46.
5. The resulting fraction is 46/88.
6. Simplify by ... |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 14^2 + 8^2 = c^2
2. Calculate the squares: 196 + 64 = c^2
3. Sum the squares: 260 = c^2
4. Take the square root of both sides: c = √260
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 16^2 = c^2
2. Calculate the squares: 121 + 256 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final A... |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^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: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + 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 +... |
Add the fractions 3/13 and 9/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 10, which is 130.
2. Convert 3/13 by multiplying top and bottom by 10: 30/130.
3. Convert 9/10 by multiplying top and bottom by 13: 117/130.
4. Add the numerators: 30 + 117 = 147.
5. The resulting fraction is 147/130.
Fina... |
Compute the quotient and remainder of 83771 divided by 65. Show the long division steps. | To divide 83771 by 65:
1. Determine how many times 65 goes into the leading digits of 83771.
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: 83771 / 65 = 1288 with a remainder of 51.
Proof: (1288 * 65... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 5^2 = c^2
2. Calculate the squares: 121 + 25 = c^2
3. Sum the squares: 146 = c^2
4. Take the square root of both sides: c = √146
Final Ans... |
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) |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 17, 16, 14, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 17 + 16 + 14 + 18) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.8, 3.2, 2.2, 0.2, 4.2]
Squared Differences: [96.04, 10.24, 4.84, 0.04, 17.64]
... |
Add the fractions 12/4 and 11/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 2, which is 4.
2. Convert 12/4 by multiplying top and bottom by 1: 12/4.
3. Convert 11/2 by multiplying top and bottom by 2: 22/4.
4. Add the numerators: 12 + 22 = 34.
5. The resulting fraction is 34/4.
6. Simplify by divid... |
Compute the indefinite integral of f(x) = 7x^1. | To find the indefinite integral ∫ 7x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 15, 19, 18, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 15 + 19 + 18 + 12) / 5 = 72 / 5 = 14.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 0.6, 4.6, 3.6, -2.4]
Squared Differences: [40.96, 0.36, 21.16, 12.96, 5.76]
... |
Compute the quotient and remainder of 91753 divided by 22. Show the long division steps. | To divide 91753 by 22:
1. Determine how many times 22 goes into the leading digits of 91753.
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: 91753 / 22 = 4170 with a remainder of 13.
Proof: (4170 * 22... |
Find the roots of the quadratic equation: 1x^2 + 0x + 3 = 0 | 1. Identify coefficients: a=1, b=0, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(1)(3) = 0 - 12 = -12.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √12i) / 2.
Final Answer: x = 0.0 ± 1.7320508075688772i |
Compute the quotient and remainder of 91184 divided by 34. Show the long division steps. | To divide 91184 by 34:
1. Determine how many times 34 goes into the leading digits of 91184.
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: 91184 / 34 = 2681 with a remainder of 30.
Proof: (2681 * 34... |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[-3, -4]
B =
[1, -1]
[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) = (-2*1) + (0*4) = -2
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (0*0) = 2
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (-4*4) = -19
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (-4*0) = 3
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 5, 6, 16, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 5 + 6 + 16 + 15) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -6.6, -5.6, 4.4, 3.4]
Squared Differences: [19.36, 43.56, 31.36, 19.36, 11.56]... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Find the roots of the quadratic equation: 3x^2 + 4x + 6 = 0 | 1. Identify coefficients: a=3, b=4, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(3)(6) = 16 - 72 = -56.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √56i) / 6.
Final Answer: x = -0.6666666666666666 ± 1.247219128924647i |
Find the roots of the quadratic equation: 6x^2 + -5x + -1 = 0 | 1. Identify coefficients: a=6, b=-5, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -24 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √49) / 12.
5. Calculate roots: x1 = 1.0, x2 = -0.16666666666666666.
Final Answer: x = 1.0,... |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Evaluate 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) |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 14, 16, 12, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 14 + 16 + 12 + 1) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, 4.4, 6.4, 2.4, -8.6]
Squared Differences: [21.16, 19.36, 40.96, 5.76, 73.96]
... |
Compute the quotient and remainder of 18904 divided by 5. Show the long division steps. | To divide 18904 by 5:
1. Determine how many times 5 goes into the leading digits of 18904.
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: 18904 / 5 = 3780 with a remainder of 4.
Proof: (3780 * 5) + 4... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Add the fractions 14/2 and 1/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 11, which is 22.
2. Convert 14/2 by multiplying top and bottom by 11: 154/22.
3. Convert 1/11 by multiplying top and bottom by 2: 2/22.
4. Add the numerators: 154 + 2 = 156.
5. The resulting fraction is 156/22.
6. Simplify ... |
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 +... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[0, -1]
B =
[0, 4]
[-1, -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) = (4*0) + (0*-1) = 0
C12 = (A11 * B12) + (A12 * B22) = (4*4) + (0*-4) = 16
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (-1*-1) = 1
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (-1*-4) = 4
Result Mat... |
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) |
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 +... |
Add the fractions 8/15 and 10/10. 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 8/15 by multiplying top and bottom by 2: 16/30.
3. Convert 10/10 by multiplying top and bottom by 3: 30/30.
4. Add the numerators: 16 + 30 = 46.
5. The resulting fraction is 46/30.
6. Simplify b... |
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 4/14 and 5/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 11, which is 154.
2. Convert 4/14 by multiplying top and bottom by 11: 44/154.
3. Convert 5/11 by multiplying top and bottom by 14: 70/154.
4. Add the numerators: 44 + 70 = 114.
5. The resulting fraction is 114/154.
6. Sim... |
Find the roots of the quadratic equation: 2x^2 + -6x + 6 = 0 | 1. Identify coefficients: a=2, b=-6, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(2)(6) = 36 - 48 = -12.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √12i) / 4.
Final Answer: x = 1.5 ± 0.8660254037844386i |
Multiply the following 2x2 matrices:
A =
[-2, 5]
[-1, 2]
B =
[0, -3]
[-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) = (-2*0) + (5*-5) = -25
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (5*-5) = -19
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (2*-5) = -10
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (2*-5) = -7
... |
Compute the quotient and remainder of 38969 divided by 61. Show the long division steps. | To divide 38969 by 61:
1. Determine how many times 61 goes into the leading digits of 38969.
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: 38969 / 61 = 638 with a remainder of 51.
Proof: (638 * 61) ... |
Multiply the following 2x2 matrices:
A =
[2, 2]
[4, -1]
B =
[-2, -1]
[-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) = (2*-2) + (2*-4) = -12
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (2*3) = 4
C21 = (A21 * B11) + (A22 * B21) = (4*-2) + (-1*-4) = -4
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (-1*3) = -7
Resul... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 19^2 = c^2
2. Calculate the squares: 16 + 361 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final Ans... |
Add the fractions 8/3 and 9/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 11, which is 33.
2. Convert 8/3 by multiplying top and bottom by 11: 88/33.
3. Convert 9/11 by multiplying top and bottom by 3: 27/33.
4. Add the numerators: 88 + 27 = 115.
5. The resulting fraction is 115/33.
Final Answer:... |
Find the derivative of f(x) = (5x + 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 = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 3x^2 + -8x + 7 = 0 | 1. Identify coefficients: a=3, b=-8, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(3)(7) = 64 - 84 = -20.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √20i) / 6.
Final Answer: x = 1.3333333333333333 ± 0.7453559924999299i |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^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: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
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 10935 divided by 98. Show the long division steps. | To divide 10935 by 98:
1. Determine how many times 98 goes into the leading digits of 10935.
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: 10935 / 98 = 111 with a remainder of 57.
Proof: (111 * 98) ... |
Add the fractions 1/5 and 11/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 2, which is 10.
2. Convert 1/5 by multiplying top and bottom by 2: 2/10.
3. Convert 11/2 by multiplying top and bottom by 5: 55/10.
4. Add the numerators: 2 + 55 = 57.
5. The resulting fraction is 57/10.
Final Answer: 57/10 |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 10^2 = c^2
2. Calculate the squares: 144 + 100 = c^2
3. Sum the squares: 244 = c^2
4. Take the square root of both sides: c = √244
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 11^2 = c^2
2. Calculate the squares: 361 + 121 = c^2
3. Sum the squares: 482 = c^2
4. Take the square root of both sides: c = √482
Final A... |
Add the fractions 4/4 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 8, which is 8.
2. Convert 4/4 by multiplying top and bottom by 2: 8/8.
3. Convert 3/8 by multiplying top and bottom by 1: 3/8.
4. Add the numerators: 8 + 3 = 11.
5. The resulting fraction is 11/8.
Final Answer: 11/8 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 7^2 = c^2
2. Calculate the squares: 121 + 49 = c^2
3. Sum the squares: 170 = c^2
4. Take the square root of both sides: c = √170
Final Ans... |
Compute the quotient and remainder of 53807 divided by 86. Show the long division steps. | To divide 53807 by 86:
1. Determine how many times 86 goes into the leading digits of 53807.
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: 53807 / 86 = 625 with a remainder of 57.
Proof: (625 * 86) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 9, 5, 13, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 9 + 5 + 13 + 11) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.2, -0.8, -4.8, 3.2, 1.2]
Squared Differences: [1.44, 0.64, 23.04, 10.24, 1.44]
... |
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... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[3, 0]
B =
[5, -1]
[-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) = (5*5) + (2*-5) = 15
C12 = (A11 * B12) + (A12 * B22) = (5*-1) + (2*0) = -5
C21 = (A21 * B11) + (A22 * B21) = (3*5) + (0*-5) = 15
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (0*0) = -3
Result Ma... |
Find the roots of the quadratic equation: 5x^2 + 0x + -9 = 0 | 1. Identify coefficients: a=5, b=0, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -180 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √180) / 10.
5. Calculate roots: x1 = 1.3416407864998738, x2 = -1.3416407864998738.
Final A... |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 10x^2 + -5x + -7 = 0 | 1. Identify coefficients: a=10, b=-5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -280 = 305.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √305) / 20.
5. Calculate roots: x1 = 1.123212459828649, x2 = -0.6232124598286489.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 6. 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 + 6^2 = c^2
2. Calculate the squares: 169 + 36 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Ans... |
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 |
Compute the quotient and remainder of 79197 divided by 38. Show the long division steps. | To divide 79197 by 38:
1. Determine how many times 38 goes into the leading digits of 79197.
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: 79197 / 38 = 2084 with a remainder of 5.
Proof: (2084 * 38)... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Add the fractions 9/7 and 10/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 11, which is 77.
2. Convert 9/7 by multiplying top and bottom by 11: 99/77.
3. Convert 10/11 by multiplying top and bottom by 7: 70/77.
4. Add the numerators: 99 + 70 = 169.
5. The resulting fraction is 169/77.
Final Answer... |
Add the fractions 4/8 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 4/8 by multiplying top and bottom by 1: 4/8.
3. Convert 3/8 by multiplying top and bottom by 1: 3/8.
4. Add the numerators: 4 + 3 = 7.
5. The resulting fraction is 7/8.
Final Answer: 7/8 |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 3^2 = c^2
2. Calculate the squares: 361 + 9 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[5, 0]
B =
[-5, -3]
[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*-5) + (-2*3) = 19
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (-2*1) = 13
C21 = (A21 * B11) + (A22 * B21) = (5*-5) + (0*3) = -25
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (0*1) = -15
Res... |
Compute the quotient and remainder of 64236 divided by 99. Show the long division steps. | To divide 64236 by 99:
1. Determine how many times 99 goes into the leading digits of 64236.
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: 64236 / 99 = 648 with a remainder of 84.
Proof: (648 * 99) ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 4^2 = c^2
2. Calculate the squares: 9 + 16 = c^2
3. Sum the squares: 25 = c^2
4. Take the square root of both sides: c = √25
Final Answer: ... |
Find the roots of the quadratic equation: 10x^2 + -7x + -7 = 0 | 1. Identify coefficients: a=10, b=-7, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -280 = 329.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √329) / 20.
5. Calculate roots: x1 = 1.2569178573608526, x2 = -0.5569178573608526.
Fina... |
Compute the quotient and remainder of 18230 divided by 83. Show the long division steps. | To divide 18230 by 83:
1. Determine how many times 83 goes into the leading digits of 18230.
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: 18230 / 83 = 219 with a remainder of 53.
Proof: (219 * 83) ... |
Find the roots of the quadratic equation: 5x^2 + 5x + 7 = 0 | 1. Identify coefficients: a=5, b=5, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(5)(7) = 25 - 140 = -115.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √115i) / 10.
Final Answer: x = -0.5 ± 1.0723805294763609i |
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 |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 8, 6, 11, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 8 + 6 + 11 + 10) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, -2.2, -4.2, 0.8, -0.2]
Squared Differences: [33.64, 4.84, 17.64, 0.64, 0.04]
... |
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 |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^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: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
Find the derivative of f(x) = (5x + 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 = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 6x^2 + 5x + 5 = 0 | 1. Identify coefficients: a=6, b=5, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(6)(5) = 25 - 120 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √95i) / 12.
Final Answer: x = -0.4166666666666667 ± 0.8122328620674136i |
Compute the quotient and remainder of 17078 divided by 83. Show the long division steps. | To divide 17078 by 83:
1. Determine how many times 83 goes into the leading digits of 17078.
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: 17078 / 83 = 205 with a remainder of 63.
Proof: (205 * 83) ... |
Add the fractions 3/10 and 5/9. 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 3/10 by multiplying top and bottom by 9: 27/90.
3. Convert 5/9 by multiplying top and bottom by 10: 50/90.
4. Add the numerators: 27 + 50 = 77.
5. The resulting fraction is 77/90.
Final Answer: 7... |
Find the derivative of f(x) = (5x + 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 = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 6x^2 + 7x + -4 = 0 | 1. Identify coefficients: a=6, b=7, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -96 = 145.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √145) / 12.
5. Calculate roots: x1 = 0.42013288156602463, x2 = -1.5867995482326913.
Final... |
Find the roots of the quadratic equation: 8x^2 + -7x + -1 = 0 | 1. Identify coefficients: a=8, b=-7, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -32 = 81.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √81) / 16.
5. Calculate roots: x1 = 1.0, x2 = -0.125.
Final Answer: x = 1.0, -0.125 |
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... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-2, -2]
B =
[0, -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*0) + (3*1) = 3
C12 = (A11 * B12) + (A12 * B22) = (-1*-3) + (3*-5) = -12
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-2*1) = -2
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-2*-5) = 16
Res... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.