prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-3, 3]
[0, -3]
B =
[-4, -1]
[0, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-4) + (3*0) = 12
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (3*-2) = -3
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (-3*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-1) + (-3*-2) = 6
Resul... |
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 |
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 |
Compute the quotient and remainder of 46373 divided by 62. Show the long division steps. | To divide 46373 by 62:
1. Determine how many times 62 goes into the leading digits of 46373.
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: 46373 / 62 = 747 with a remainder of 59.
Proof: (747 * 62) ... |
Add the fractions 1/8 and 10/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 13, which is 104.
2. Convert 1/8 by multiplying top and bottom by 13: 13/104.
3. Convert 10/13 by multiplying top and bottom by 8: 80/104.
4. Add the numerators: 13 + 80 = 93.
5. The resulting fraction is 93/104.
Final Answ... |
Add the fractions 2/15 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 13, which is 195.
2. Convert 2/15 by multiplying top and bottom by 13: 26/195.
3. Convert 6/13 by multiplying top and bottom by 15: 90/195.
4. Add the numerators: 26 + 90 = 116.
5. The resulting fraction is 116/195.
Final ... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 18, 2, 10, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 18 + 2 + 10 + 20) / 5 = 70 / 5 = 14.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.0, 4.0, -12.0, -4.0, 6.0]
Squared Differences: [36.0, 16.0, 144.0, 16.0, 36.0]
... |
Find the roots of the quadratic equation: 8x^2 + -3x + -8 = 0 | 1. Identify coefficients: a=8, b=-3, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -256 = 265.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √265) / 16.
5. Calculate roots: x1 = 1.2049262872562316, x2 = -0.8299262872562316.
Final ... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (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 +... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 19^2 = c^2
2. Calculate the squares: 400 + 361 = c^2
3. Sum the squares: 761 = c^2
4. Take the square root of both sides: c = √761
Final A... |
Add the fractions 6/12 and 4/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 7, which is 84.
2. Convert 6/12 by multiplying top and bottom by 7: 42/84.
3. Convert 4/7 by multiplying top and bottom by 12: 48/84.
4. Add the numerators: 42 + 48 = 90.
5. The resulting fraction is 90/84.
6. Simplify by ... |
Find the roots of the quadratic equation: 10x^2 + 10x + -1 = 0 | 1. Identify coefficients: a=10, b=10, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -40 = 140.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √140) / 20.
5. Calculate roots: x1 = 0.09160797830996162, x2 = -1.0916079783099615.
F... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-3, -3]
B =
[2, -2]
[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) = (2*2) + (3*4) = 16
C12 = (A11 * B12) + (A12 * B22) = (2*-2) + (3*-5) = -19
C21 = (A21 * B11) + (A22 * B21) = (-3*2) + (-3*4) = -18
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (-3*-5) = 21
Res... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 20^2 = c^2
2. Calculate the squares: 324 + 400 = c^2
3. Sum the squares: 724 = c^2
4. Take the square root of both sides: c = √724
Final A... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[-5, 4]
B =
[5, 4]
[4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*5) + (-2*4) = -3
C12 = (A11 * B12) + (A12 * B22) = (1*4) + (-2*5) = -6
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (4*4) = -9
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (4*5) = 0
Result Mat... |
Find the roots of the quadratic equation: 4x^2 + 4x + -10 = 0 | 1. Identify coefficients: a=4, b=4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -160 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √176) / 8.
5. Calculate roots: x1 = 1.1583123951777, x2 = -2.1583123951777.
Final Answe... |
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 +... |
Find the roots of the quadratic equation: 2x^2 + -7x + 1 = 0 | 1. Identify coefficients: a=2, b=-7, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 8 = 41.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √41) / 4.
5. Calculate roots: x1 = 3.350781059358212, x2 = 0.14921894064178787.
Final Answer:... |
Compute the quotient and remainder of 69625 divided by 25. Show the long division steps. | To divide 69625 by 25:
1. Determine how many times 25 goes into the leading digits of 69625.
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: 69625 / 25 = 2785 with a remainder of 0.
Proof: (2785 * 25)... |
Find the roots of the quadratic equation: 7x^2 + 5x + -10 = 0 | 1. Identify coefficients: a=7, b=5, c=-10.
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) / 14.
5. Calculate roots: x1 = 0.8903035140409271, x2 = -1.6045892283266414.
Fina... |
Find the roots of the quadratic equation: 8x^2 + 1x + -2 = 0 | 1. Identify coefficients: a=8, b=1, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -64 = 65.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √65) / 16.
5. Calculate roots: x1 = 0.4413911092686593, x2 = -0.5663911092686593.
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 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 |
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 |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Find the roots of the quadratic equation: 4x^2 + -8x + -8 = 0 | 1. Identify coefficients: a=4, b=-8, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -128 = 192.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √192) / 8.
5. Calculate roots: x1 = 2.732050807568877, x2 = -0.7320508075688772.
Final A... |
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 |
Add the fractions 9/7 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 13, which is 91.
2. Convert 9/7 by multiplying top and bottom by 13: 117/91.
3. Convert 6/13 by multiplying top and bottom by 7: 42/91.
4. Add the numerators: 117 + 42 = 159.
5. The resulting fraction is 159/91.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[-4, -3]
[3, -2]
B =
[1, -2]
[5, 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*1) + (-3*5) = -19
C12 = (A11 * B12) + (A12 * B22) = (-4*-2) + (-3*4) = -4
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-2*5) = -7
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-2*4) = -14
Res... |
Find the roots of the quadratic equation: 7x^2 + 10x + -10 = 0 | 1. Identify coefficients: a=7, b=10, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -280 = 380.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √380) / 14.
5. Calculate roots: x1 = 0.6781134778298519, x2 = -2.1066849064012803.
F... |
Find the roots of the quadratic equation: 5x^2 + -5x + -10 = 0 | 1. Identify coefficients: a=5, b=-5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -200 = 225.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √225) / 10.
5. Calculate roots: x1 = 2.0, x2 = -1.0.
Final Answer: x = 2.0, -1.0 |
Multiply the following 2x2 matrices:
A =
[-4, 4]
[0, -5]
B =
[5, 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) = (-4*5) + (4*1) = -16
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (4*2) = 8
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (-5*1) = -5
C22 = (A21 * B12) + (A22 * B22) = (0*0) + (-5*2) = -10
Result M... |
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) |
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 ... |
Add the fractions 11/12 and 8/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 8, which is 24.
2. Convert 11/12 by multiplying top and bottom by 2: 22/24.
3. Convert 8/8 by multiplying top and bottom by 3: 24/24.
4. Add the numerators: 22 + 24 = 46.
5. The resulting fraction is 46/24.
6. Simplify by ... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
Compute the quotient and remainder of 63631 divided by 86. Show the long division steps. | To divide 63631 by 86:
1. Determine how many times 86 goes into the leading digits of 63631.
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: 63631 / 86 = 739 with a remainder of 77.
Proof: (739 * 86) ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 7^2 = c^2
2. Calculate the squares: 196 + 49 = c^2
3. Sum the squares: 245 = c^2
4. Take the square root of both sides: c = √245
Final Ans... |
Find the roots of the quadratic equation: 6x^2 + 1x + -10 = 0 | 1. Identify coefficients: a=6, b=1, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -240 = 241.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √241) / 12.
5. Calculate roots: x1 = 1.210347891355002, x2 = -1.3770145580216688.
Final ... |
Find the roots of the quadratic equation: 8x^2 + -1x + -6 = 0 | 1. Identify coefficients: a=8, b=-1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -192 = 193.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √193) / 16.
5. Calculate roots: x1 = 0.9307777493406127, x2 = -0.8057777493406127.
Final ... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 12, 15, 20, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 12 + 15 + 20 + 4) / 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, 0.0, 3.0, 8.0, -8.0]
Squared Differences: [9.0, 0.0, 9.0, 64.0, 64.0]
Sum ... |
Add the fractions 11/9 and 14/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 12, which is 36.
2. Convert 11/9 by multiplying top and bottom by 4: 44/36.
3. Convert 14/12 by multiplying top and bottom by 3: 42/36.
4. Add the numerators: 44 + 42 = 86.
5. The resulting fraction is 86/36.
6. Simplify by... |
Compute the quotient and remainder of 67061 divided by 68. Show the long division steps. | To divide 67061 by 68:
1. Determine how many times 68 goes into the leading digits of 67061.
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: 67061 / 68 = 986 with a remainder of 13.
Proof: (986 * 68) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 8, 19, 5, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 8 + 19 + 5 + 20) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.8, -4.8, 6.2, -7.8, 7.2]
Squared Differences: [0.64, 23.04, 38.44, 60.84, 51.84]... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (4x + 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 = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 6^2 = c^2
2. Calculate the squares: 36 + 36 = c^2
3. Sum the squares: 72 = c^2
4. Take the square root of both sides: c = √72
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[3, 0]
[-2, 1]
B =
[-3, -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) = (3*-3) + (0*-5) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*-1) + (0*2) = -3
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (1*-5) = 1
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (1*2) = 4
Result ... |
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) |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 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 |
Add the fractions 10/15 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 3, which is 15.
2. Convert 10/15 by multiplying top and bottom by 1: 10/15.
3. Convert 14/3 by multiplying top and bottom by 5: 70/15.
4. Add the numerators: 10 + 70 = 80.
5. The resulting fraction is 80/15.
6. Simplify by... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 17, 20, 19, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 17 + 20 + 19 + 8) / 5 = 84 / 5 = 16.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, 0.2, 3.2, 2.2, -8.8]
Squared Differences: [10.24, 0.04, 10.24, 4.84, 77.44]
... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Add the fractions 6/5 and 9/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 13, which is 65.
2. Convert 6/5 by multiplying top and bottom by 13: 78/65.
3. Convert 9/13 by multiplying top and bottom by 5: 45/65.
4. Add the numerators: 78 + 45 = 123.
5. The resulting fraction is 123/65.
Final Answer:... |
Compute the quotient and remainder of 35951 divided by 16. Show the long division steps. | To divide 35951 by 16:
1. Determine how many times 16 goes into the leading digits of 35951.
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: 35951 / 16 = 2246 with a remainder of 15.
Proof: (2246 * 16... |
Add the fractions 15/2 and 8/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 15/2 by multiplying top and bottom by 7: 105/14.
3. Convert 8/14 by multiplying top and bottom by 1: 8/14.
4. Add the numerators: 105 + 8 = 113.
5. The resulting fraction is 113/14.
Final Answer:... |
Add the fractions 2/11 and 9/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 2/11 by multiplying top and bottom by 7: 14/77.
3. Convert 9/7 by multiplying top and bottom by 11: 99/77.
4. Add the numerators: 14 + 99 = 113.
5. The resulting fraction is 113/77.
Final Answer:... |
Find the roots of the quadratic equation: 9x^2 + -1x + 2 = 0 | 1. Identify coefficients: a=9, b=-1, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(9)(2) = 1 - 72 = -71.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √71i) / 18.
Final Answer: x = 0.05555555555555555 ± 0.46811943184313104i |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 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 |
Compute the quotient and remainder of 51490 divided by 77. Show the long division steps. | To divide 51490 by 77:
1. Determine how many times 77 goes into the leading digits of 51490.
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: 51490 / 77 = 668 with a remainder of 54.
Proof: (668 * 77) ... |
Compute the quotient and remainder of 48123 divided by 98. Show the long division steps. | To divide 48123 by 98:
1. Determine how many times 98 goes into the leading digits of 48123.
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: 48123 / 98 = 491 with a remainder of 5.
Proof: (491 * 98) +... |
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) |
Add the fractions 7/6 and 15/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 8, which is 24.
2. Convert 7/6 by multiplying top and bottom by 4: 28/24.
3. Convert 15/8 by multiplying top and bottom by 3: 45/24.
4. Add the numerators: 28 + 45 = 73.
5. The resulting fraction is 73/24.
Final Answer: 73/... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 9^2 = c^2
2. Calculate the squares: 144 + 81 = c^2
3. Sum the squares: 225 = c^2
4. Take the square root of both sides: c = √225
Final Ans... |
Compute the quotient and remainder of 43138 divided by 2. Show the long division steps. | To divide 43138 by 2:
1. Determine how many times 2 goes into the leading digits of 43138.
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: 43138 / 2 = 21569 with a remainder of 0.
Proof: (21569 * 2) +... |
Add the fractions 6/2 and 3/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 12, which is 12.
2. Convert 6/2 by multiplying top and bottom by 6: 36/12.
3. Convert 3/12 by multiplying top and bottom by 1: 3/12.
4. Add the numerators: 36 + 3 = 39.
5. The resulting fraction is 39/12.
6. Simplify by div... |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^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: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
Find the roots of the quadratic equation: 4x^2 + 4x + 2 = 0 | 1. Identify coefficients: a=4, b=4, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(4)(2) = 16 - 32 = -16.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √16i) / 8.
Final Answer: x = -0.5 ± 0.5i |
Find the roots of the quadratic equation: 2x^2 + -2x + 1 = 0 | 1. Identify coefficients: a=2, b=-2, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(2)(1) = 4 - 8 = -4.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √4i) / 4.
Final Answer: x = 0.5 ± 0.5i |
Add the fractions 12/11 and 10/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 8, which is 88.
2. Convert 12/11 by multiplying top and bottom by 8: 96/88.
3. Convert 10/8 by multiplying top and bottom by 11: 110/88.
4. Add the numerators: 96 + 110 = 206.
5. The resulting fraction is 206/88.
6. Simpli... |
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 |
Compute the quotient and remainder of 75504 divided by 24. Show the long division steps. | To divide 75504 by 24:
1. Determine how many times 24 goes into the leading digits of 75504.
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: 75504 / 24 = 3146 with a remainder of 0.
Proof: (3146 * 24)... |
Add the fractions 15/12 and 11/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 12, which is 12.
2. Convert 15/12 by multiplying top and bottom by 1: 15/12.
3. Convert 11/12 by multiplying top and bottom by 1: 11/12.
4. Add the numerators: 15 + 11 = 26.
5. The resulting fraction is 26/12.
6. Simplify ... |
Compute the quotient and remainder of 88520 divided by 58. Show the long division steps. | To divide 88520 by 58:
1. Determine how many times 58 goes into the leading digits of 88520.
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: 88520 / 58 = 1526 with a remainder of 12.
Proof: (1526 * 58... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 17^2 = c^2
2. Calculate the squares: 64 + 289 = c^2
3. Sum the squares: 353 = c^2
4. Take the square root of both sides: c = √353
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 19, 3, 5, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 19 + 3 + 5 + 10) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, 9.6, -6.4, -4.4, 0.6]
Squared Differences: [0.36, 92.16, 40.96, 19.36, 0.36]
... |
Find the roots of the quadratic equation: 2x^2 + 3x + -10 = 0 | 1. Identify coefficients: a=2, b=3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -80 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √89) / 4.
5. Calculate roots: x1 = 1.6084952830141508, x2 = -3.108495283014151.
Final Answ... |
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 +... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 12, 19, 4, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 12 + 19 + 4 + 14) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, 0.0, 7.0, -8.0, 2.0]
Squared Differences: [1.0, 0.0, 49.0, 64.0, 4.0]
Sum... |
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 tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 4512 divided by 92. Show the long division steps. | To divide 4512 by 92:
1. Determine how many times 92 goes into the leading digits of 4512.
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: 4512 / 92 = 49 with a remainder of 4.
Proof: (49 * 92) + 4 = ... |
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) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 17, 6, 11, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 17 + 6 + 11 + 4) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, 7.8, -3.2, 1.8, -5.2]
Squared Differences: [1.44, 60.84, 10.24, 3.24, 27.04]
... |
Compute the quotient and remainder of 40961 divided by 16. Show the long division steps. | To divide 40961 by 16:
1. Determine how many times 16 goes into the leading digits of 40961.
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: 40961 / 16 = 2560 with a remainder of 1.
Proof: (2560 * 16)... |
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 |
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 roots of the quadratic equation: 1x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=1, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -4 = 68.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √68) / 2.
5. Calculate roots: x1 = 8.123105625617661, x2 = -0.12310562561766059.
Final Answ... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 6, 9, 17, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 6 + 9 + 17 + 4) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.4, -3.6, -0.6, 7.4, -5.6]
Squared Differences: [5.76, 12.96, 0.36, 54.76, 31.36]
... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 15^2 = c^2
2. Calculate the squares: 16 + 225 = c^2
3. Sum the squares: 241 = c^2
4. Take the square root of both sides: c = √241
Final Ans... |
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