prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Compute the quotient and remainder of 48168 divided by 33. Show the long division steps. | To divide 48168 by 33:
1. Determine how many times 33 goes into the leading digits of 48168.
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: 48168 / 33 = 1459 with a remainder of 21.
Proof: (1459 * 33... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 96805 divided by 33. Show the long division steps. | To divide 96805 by 33:
1. Determine how many times 33 goes into the leading digits of 96805.
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: 96805 / 33 = 2933 with a remainder of 16.
Proof: (2933 * 33... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 5, 10, 19, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 5 + 10 + 19 + 12) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.0, -8.0, -3.0, 6.0, -1.0]
Squared Differences: [36.0, 64.0, 9.0, 36.0, 1.0]
S... |
Compute the quotient and remainder of 45568 divided by 27. Show the long division steps. | To divide 45568 by 27:
1. Determine how many times 27 goes into the leading digits of 45568.
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: 45568 / 27 = 1687 with a remainder of 19.
Proof: (1687 * 27... |
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... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 18, 2, 11, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 18 + 2 + 11 + 16) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, 6.2, -9.8, -0.8, 4.2]
Squared Differences: [0.04, 38.44, 96.04, 0.64, 17.64]
... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 19^2 = c^2
2. Calculate the squares: 225 + 361 = c^2
3. Sum the squares: 586 = c^2
4. Take the square root of both sides: c = √586
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 15, 2, 9, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 15 + 2 + 9 + 1) / 5 = 29 / 5 = 5.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, 9.2, -3.8, 3.2, -4.8]
Squared Differences: [14.44, 84.64, 14.44, 10.24, 23.04]
... |
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 +... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 38355 divided by 36. Show the long division steps. | To divide 38355 by 36:
1. Determine how many times 36 goes into the leading digits of 38355.
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: 38355 / 36 = 1065 with a remainder of 15.
Proof: (1065 * 36... |
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: 9x^2 + -3x + 7 = 0 | 1. Identify coefficients: a=9, b=-3, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(9)(7) = 9 - 252 = -243.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √243i) / 18.
Final Answer: x = 0.16666666666666666 ± 0.8660254037844387i |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[-3, 3]
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) + (-4*-5) = 15
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (-4*3) = -8
C21 = (A21 * B11) + (A22 * B21) = (-3*5) + (3*-5) = -30
C22 = (A21 * B12) + (A22 * B22) = (-3*-4) + (3*3) = 21
Re... |
Multiply the following 2x2 matrices:
A =
[-2, -2]
[-1, 5]
B =
[-3, 3]
[5, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-3) + (-2*5) = -4
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (-2*-1) = -4
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (5*5) = 28
C22 = (A21 * B12) + (A22 * B22) = (-1*3) + (5*-1) = -8
Res... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 7^2 = c^2
2. Calculate the squares: 144 + 49 = c^2
3. Sum the squares: 193 = c^2
4. Take the square root of both sides: c = √193
Final Ans... |
Compute the quotient and remainder of 21554 divided by 76. Show the long division steps. | To divide 21554 by 76:
1. Determine how many times 76 goes into the leading digits of 21554.
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: 21554 / 76 = 283 with a remainder of 46.
Proof: (283 * 76) ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 12, 5, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 12 + 5 + 9 + 18) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 2.8, -4.2, -0.2, 8.8]
Squared Differences: [51.84, 7.84, 17.64, 0.04, 77.44]
... |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^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: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Find the roots of the quadratic equation: 3x^2 + 5x + -2 = 0 | 1. Identify coefficients: a=3, b=5, c=-2.
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) / 6.
5. Calculate roots: x1 = 0.3333333333333333, x2 = -2.0.
Final Answer: x = 0.3333... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
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 |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[-1, 5]
B =
[-3, -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) = (-4*-3) + (-1*-5) = 17
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-1*-5) = 17
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (5*-5) = -22
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (5*-5) = -2... |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[-5, -5]
B =
[-3, -4]
[-1, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-3) + (0*-1) = 6
C12 = (A11 * B12) + (A12 * B22) = (-2*-4) + (0*1) = 8
C21 = (A21 * B11) + (A22 * B21) = (-5*-3) + (-5*-1) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (-5*1) = 15
Res... |
Add the fractions 13/3 and 8/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 7, which is 21.
2. Convert 13/3 by multiplying top and bottom by 7: 91/21.
3. Convert 8/7 by multiplying top and bottom by 3: 24/21.
4. Add the numerators: 91 + 24 = 115.
5. The resulting fraction is 115/21.
Final Answer: 1... |
Add the fractions 7/4 and 11/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 7/4 by multiplying top and bottom by 9: 63/36.
3. Convert 11/9 by multiplying top and bottom by 4: 44/36.
4. Add the numerators: 63 + 44 = 107.
5. The resulting fraction is 107/36.
Final Answer: 1... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 4^2 = c^2
2. Calculate the squares: 16 + 16 = c^2
3. Sum the squares: 32 = c^2
4. Take the square root of both sides: c = √32
Final Answer:... |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^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: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + 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 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) |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 3^2 = c^2
2. Calculate the squares: 196 + 9 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Answ... |
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 + 7x + -5 = 0 | 1. Identify coefficients: a=2, b=7, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -40 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √89) / 4.
5. Calculate roots: x1 = 0.6084952830141508, x2 = -4.10849528301415.
Final Answe... |
Find the roots of the quadratic equation: 8x^2 + 3x + -10 = 0 | 1. Identify coefficients: a=8, b=3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -320 = 329.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √329) / 16.
5. Calculate roots: x1 = 0.9461473217010659, x2 = -1.3211473217010659.
Final... |
Find the roots of the quadratic equation: 5x^2 + -1x + 0 = 0 | 1. Identify coefficients: a=5, b=-1, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - 0 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √1) / 10.
5. Calculate roots: x1 = 0.2, x2 = 0.0.
Final Answer: x = 0.2, 0.0 |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[-3, 5]
B =
[1, -2]
[2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*1) + (-5*2) = -14
C12 = (A11 * B12) + (A12 * B22) = (-4*-2) + (-5*0) = 8
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (5*2) = 7
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (5*0) = 6
Result ... |
Multiply the following 2x2 matrices:
A =
[-5, -4]
[-2, 4]
B =
[1, 3]
[-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) = (-5*1) + (-4*-4) = 11
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (-4*5) = -35
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (4*-4) = -18
C22 = (A21 * B12) + (A22 * B22) = (-2*3) + (4*5) = 14
Res... |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[-4, -3]
B =
[-5, -3]
[-1, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-5) + (-4*-1) = 9
C12 = (A11 * B12) + (A12 * B22) = (-1*-3) + (-4*-1) = 7
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (-3*-1) = 23
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (-3*-1) = 15
... |
Find the roots of the quadratic equation: 8x^2 + -7x + 9 = 0 | 1. Identify coefficients: a=8, b=-7, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(8)(9) = 49 - 288 = -239.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √239i) / 16.
Final Answer: x = 0.4375 ± 0.9662265521087692i |
Multiply the following 2x2 matrices:
A =
[4, -3]
[-5, 0]
B =
[-5, -3]
[0, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-5) + (-3*0) = -20
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (-3*5) = -27
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (0*0) = 25
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (0*5) = 15
Res... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 10^2 = c^2
2. Calculate the squares: 64 + 100 = c^2
3. Sum the squares: 164 = c^2
4. Take the square root of both sides: c = √164
Final Ans... |
Add the fractions 5/8 and 2/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 5, which is 40.
2. Convert 5/8 by multiplying top and bottom by 5: 25/40.
3. Convert 2/5 by multiplying top and bottom by 8: 16/40.
4. Add the numerators: 25 + 16 = 41.
5. The resulting fraction is 41/40.
Final Answer: 41/4... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[1, -3]
B =
[2, 3]
[2, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*2) + (1*2) = -4
C12 = (A11 * B12) + (A12 * B22) = (-3*3) + (1*-2) = -11
C21 = (A21 * B11) + (A22 * B21) = (1*2) + (-3*2) = -4
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-3*-2) = 9
Result ... |
Find the roots of the quadratic equation: 10x^2 + -3x + 6 = 0 | 1. Identify coefficients: a=10, b=-3, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(10)(6) = 9 - 240 = -231.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √231i) / 20.
Final Answer: x = 0.15 ± 0.7599342076785331i |
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 indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 6481 divided by 57. Show the long division steps. | To divide 6481 by 57:
1. Determine how many times 57 goes into the leading digits of 6481.
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: 6481 / 57 = 113 with a remainder of 40.
Proof: (113 * 57) + 4... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 8, 13, 3, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 8 + 13 + 3 + 8) / 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, -2.4, 2.6, -7.4, -2.4]
Squared Differences: [92.16, 5.76, 6.76, 54.76, 5.76]
... |
Multiply the following 2x2 matrices:
A =
[5, -3]
[-4, 1]
B =
[1, -2]
[1, 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*1) + (-3*1) = 2
C12 = (A11 * B12) + (A12 * B22) = (5*-2) + (-3*0) = -10
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (1*1) = -3
C22 = (A21 * B12) + (A22 * B22) = (-4*-2) + (1*0) = 8
Result M... |
Evaluate the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 8x^2 + 6x + 4 = 0 | 1. Identify coefficients: a=8, b=6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(8)(4) = 36 - 128 = -92.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √92i) / 16.
Final Answer: x = -0.375 ± 0.5994789404140899i |
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 |
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 ... |
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) |
Find the roots of the quadratic equation: 8x^2 + 7x + -5 = 0 | 1. Identify coefficients: a=8, b=7, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -160 = 209.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √209) / 16.
5. Calculate roots: x1 = 0.46605201842506006, x2 = -1.34105201842506.
Final ... |
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 ... |
Multiply the following 2x2 matrices:
A =
[0, -2]
[-4, 2]
B =
[-3, 5]
[0, 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*-3) + (-2*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (-2*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-4*-3) + (2*0) = 12
C22 = (A21 * B12) + (A22 * B22) = (-4*5) + (2*0) = -20
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 20. 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 + 20^2 = c^2
2. Calculate the squares: 81 + 400 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final Ans... |
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 |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 40551 divided by 45. Show the long division steps. | To divide 40551 by 45:
1. Determine how many times 45 goes into the leading digits of 40551.
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: 40551 / 45 = 901 with a remainder of 6.
Proof: (901 * 45) +... |
Compute the quotient and remainder of 93867 divided by 13. Show the long division steps. | To divide 93867 by 13:
1. Determine how many times 13 goes into the leading digits of 93867.
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: 93867 / 13 = 7220 with a remainder of 7.
Proof: (7220 * 13)... |
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) |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
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 |
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 =
[4, -4]
[1, -1]
B =
[-5, 1]
[0, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-5) + (-4*0) = -20
C12 = (A11 * B12) + (A12 * B22) = (4*1) + (-4*-3) = 16
C21 = (A21 * B11) + (A22 * B21) = (1*-5) + (-1*0) = -5
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (-1*-3) = 4
Resul... |
Add the fractions 14/5 and 8/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 4, which is 20.
2. Convert 14/5 by multiplying top and bottom by 4: 56/20.
3. Convert 8/4 by multiplying top and bottom by 5: 40/20.
4. Add the numerators: 56 + 40 = 96.
5. The resulting fraction is 96/20.
6. Simplify by di... |
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 derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Compute the 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 |
Add the fractions 9/14 and 11/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 9/14 by multiplying top and bottom by 3: 27/42.
3. Convert 11/3 by multiplying top and bottom by 14: 154/42.
4. Add the numerators: 27 + 154 = 181.
5. The resulting fraction is 181/42.
Final Answ... |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[3, -1]
[5, -1]
B =
[-5, 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*-5) + (-1*4) = -19
C12 = (A11 * B12) + (A12 * B22) = (3*3) + (-1*4) = 5
C21 = (A21 * B11) + (A22 * B21) = (5*-5) + (-1*4) = -29
C22 = (A21 * B12) + (A22 * B22) = (5*3) + (-1*4) = 11
Result... |
Add the fractions 9/15 and 1/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 15, which is 15.
2. Convert 9/15 by multiplying top and bottom by 1: 9/15.
3. Convert 1/15 by multiplying top and bottom by 1: 1/15.
4. Add the numerators: 9 + 1 = 10.
5. The resulting fraction is 10/15.
6. Simplify by div... |
Compute the indefinite integral of f(x) = 8x^2. | To find the indefinite integral ∫ 8x^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: 8 / 3 = 8/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^2 dx = 8/3x^3 + 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 +... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 18^2 = c^2
2. Calculate the squares: 144 + 324 = c^2
3. Sum the squares: 468 = c^2
4. Take the square root of both sides: c = √468
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 12, 14, 14, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 12 + 14 + 14 + 13) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.0, 0.0, 2.0, 2.0, 1.0]
Squared Differences: [25.0, 0.0, 4.0, 4.0, 1.0]
Sum o... |
Add the fractions 6/10 and 15/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 2, which is 10.
2. Convert 6/10 by multiplying top and bottom by 1: 6/10.
3. Convert 15/2 by multiplying top and bottom by 5: 75/10.
4. Add the numerators: 6 + 75 = 81.
5. The resulting fraction is 81/10.
Final Answer: 81/... |
Compute the indefinite integral of f(x) = 3x^4. | To find the indefinite integral ∫ 3x^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: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[-2, -1]
[2, -5]
B =
[1, -3]
[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) = (-2*1) + (-1*5) = -7
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (-1*-2) = 8
C21 = (A21 * B11) + (A22 * B21) = (2*1) + (-5*5) = -23
C22 = (A21 * B12) + (A22 * B22) = (2*-3) + (-5*-2) = 4
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 13, 15, 15, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 13 + 15 + 15 + 16) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 0.0, 2.0, 2.0, 3.0]
Squared Differences: [49.0, 0.0, 4.0, 4.0, 9.0]
Sum o... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 6, 17, 13, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 6 + 17 + 13 + 17) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, -7.6, 3.4, -0.6, 3.4]
Squared Differences: [1.96, 57.76, 11.56, 0.36, 11.56]
... |
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 |
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 |
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 +... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_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 =
[1, 1]
[-3, 4]
B =
[-3, 0]
[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) = (1*-3) + (1*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (1*-2) = -2
C21 = (A21 * B11) + (A22 * B21) = (-3*-3) + (4*1) = 13
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (4*-2) = -8
Result ... |
Find the roots of the quadratic equation: 3x^2 + 6x + 8 = 0 | 1. Identify coefficients: a=3, b=6, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(3)(8) = 36 - 96 = -60.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √60i) / 6.
Final Answer: x = -1.0 ± 1.2909944487358056i |
Find the roots of the quadratic equation: 6x^2 + -8x + -8 = 0 | 1. Identify coefficients: a=6, b=-8, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -192 = 256.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √256) / 12.
5. Calculate roots: x1 = 2.0, x2 = -0.6666666666666666.
Final Answer: x = 2.... |
Add the fractions 10/12 and 7/4. 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 10/12 by multiplying top and bottom by 1: 10/12.
3. Convert 7/4 by multiplying top and bottom by 3: 21/12.
4. Add the numerators: 10 + 21 = 31.
5. The resulting fraction is 31/12.
Final Answer: 3... |
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... |
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) |
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 ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.