prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-5, -3]
[-2, 4]
B =
[4, -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) = (-5*4) + (-3*-5) = -5
C12 = (A11 * B12) + (A12 * B22) = (-5*-4) + (-3*3) = 11
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (4*-5) = -28
C22 = (A21 * B12) + (A22 * B22) = (-2*-4) + (4*3) = 20
Re... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 16^2 = c^2
2. Calculate the squares: 9 + 256 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 14, 8, 5, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 14 + 8 + 5 + 8) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, 6.4, 0.4, -2.6, 0.4]
Squared Differences: [21.16, 40.96, 0.16, 6.76, 0.16]
Su... |
Compute the quotient and remainder of 90379 divided by 57. Show the long division steps. | To divide 90379 by 57:
1. Determine how many times 57 goes into the leading digits of 90379.
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: 90379 / 57 = 1585 with a remainder of 34.
Proof: (1585 * 57... |
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 |
Multiply the following 2x2 matrices:
A =
[0, -2]
[-5, 4]
B =
[-2, -1]
[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) = (0*-2) + (-2*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (-2*1) = -2
C21 = (A21 * B11) + (A22 * B21) = (-5*-2) + (4*1) = 14
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (4*1) = 9
Result... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 18^2 = c^2
2. Calculate the squares: 324 + 324 = c^2
3. Sum the squares: 648 = c^2
4. Take the square root of both sides: c = √648
Final A... |
Find the roots of the quadratic equation: 2x^2 + 0x + 4 = 0 | 1. Identify coefficients: a=2, b=0, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(2)(4) = 0 - 32 = -32.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √32i) / 4.
Final Answer: x = 0.0 ± 1.4142135623730951i |
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 roots of the quadratic equation: 10x^2 + -8x + -7 = 0 | 1. Identify coefficients: a=10, b=-8, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -280 = 344.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √344) / 20.
5. Calculate roots: x1 = 1.3273618495495705, x2 = -0.5273618495495704.
Fina... |
Add the fractions 13/3 and 11/11. 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 13/3 by multiplying top and bottom by 11: 143/33.
3. Convert 11/11 by multiplying top and bottom by 3: 33/33.
4. Add the numerators: 143 + 33 = 176.
5. The resulting fraction is 176/33.
6. Simpli... |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[1, -3]
[-2, 2]
B =
[1, 5]
[0, 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) = (1*1) + (-3*0) = 1
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (-3*4) = -7
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (2*0) = -2
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (2*4) = -2
Result Mat... |
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 ... |
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 6/2 and 2/14. 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 6/2 by multiplying top and bottom by 7: 42/14.
3. Convert 2/14 by multiplying top and bottom by 1: 2/14.
4. Add the numerators: 42 + 2 = 44.
5. The resulting fraction is 44/14.
6. Simplify by div... |
Add the fractions 10/10 and 1/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 5, which is 10.
2. Convert 10/10 by multiplying top and bottom by 1: 10/10.
3. Convert 1/5 by multiplying top and bottom by 2: 2/10.
4. Add the numerators: 10 + 2 = 12.
5. The resulting fraction is 12/10.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[1, -3]
B =
[5, 5]
[3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*5) + (-4*3) = -12
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (-4*-4) = 16
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-3*3) = -4
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (-3*-4) = 17
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 1, 8, 5, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 1 + 8 + 5 + 6) / 5 = 23 / 5 = 4.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.6, -3.6, 3.4, 0.4, 1.4]
Squared Differences: [2.56, 12.96, 11.56, 0.16, 1.96]
Sum... |
Add the fractions 7/3 and 1/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 9, which is 9.
2. Convert 7/3 by multiplying top and bottom by 3: 21/9.
3. Convert 1/9 by multiplying top and bottom by 1: 1/9.
4. Add the numerators: 21 + 1 = 22.
5. The resulting fraction is 22/9.
Final Answer: 22/9 |
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 |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 12, 12, 3, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 12 + 12 + 3 + 20) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, 1.2, 1.2, -7.8, 9.2]
Squared Differences: [14.44, 1.44, 1.44, 60.84, 84.64]
... |
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 +... |
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 10/7 and 7/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 4, which is 28.
2. Convert 10/7 by multiplying top and bottom by 4: 40/28.
3. Convert 7/4 by multiplying top and bottom by 7: 49/28.
4. Add the numerators: 40 + 49 = 89.
5. The resulting fraction is 89/28.
Final Answer: 89/... |
Compute the quotient and remainder of 57308 divided by 45. Show the long division steps. | To divide 57308 by 45:
1. Determine how many times 45 goes into the leading digits of 57308.
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: 57308 / 45 = 1273 with a remainder of 23.
Proof: (1273 * 45... |
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... |
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 |
Find the roots of the quadratic equation: 8x^2 + -10x + -8 = 0 | 1. Identify coefficients: a=8, b=-10, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -256 = 356.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √356) / 16.
5. Calculate roots: x1 = 1.8042476415070754, x2 = -0.5542476415070754.
Fi... |
Add the fractions 2/10 and 8/2. 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 2/10 by multiplying top and bottom by 1: 2/10.
3. Convert 8/2 by multiplying top and bottom by 5: 40/10.
4. Add the numerators: 2 + 40 = 42.
5. The resulting fraction is 42/10.
6. Simplify by div... |
Find the roots of the quadratic equation: 1x^2 + -4x + 5 = 0 | 1. Identify coefficients: a=1, b=-4, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(1)(5) = 16 - 20 = -4.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √4i) / 2.
Final Answer: x = 2.0 ± 1.0i |
Add the fractions 7/3 and 14/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 2, which is 6.
2. Convert 7/3 by multiplying top and bottom by 2: 14/6.
3. Convert 14/2 by multiplying top and bottom by 3: 42/6.
4. Add the numerators: 14 + 42 = 56.
5. The resulting fraction is 56/6.
6. Simplify by dividi... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 13^2 = c^2
2. Calculate the squares: 64 + 169 = c^2
3. Sum the squares: 233 = c^2
4. Take the square root of both sides: c = √233
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 19^2 = c^2
2. Calculate the squares: 324 + 361 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
Final A... |
Add the fractions 8/4 and 15/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 4, which is 4.
2. Convert 8/4 by multiplying top and bottom by 1: 8/4.
3. Convert 15/4 by multiplying top and bottom by 1: 15/4.
4. Add the numerators: 8 + 15 = 23.
5. The resulting fraction is 23/4.
Final Answer: 23/4 |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 5^2 = c^2
2. Calculate the squares: 36 + 25 = c^2
3. Sum the squares: 61 = c^2
4. Take the square root of both sides: c = √61
Final Answer:... |
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 |
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) |
Multiply the following 2x2 matrices:
A =
[0, 4]
[2, 2]
B =
[4, 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) = (0*4) + (4*-5) = -20
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (4*-4) = -16
C21 = (A21 * B11) + (A22 * B21) = (2*4) + (2*-5) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (2*-4) = -4
Result ... |
Find the roots of the quadratic equation: 6x^2 + 10x + -3 = 0 | 1. Identify coefficients: a=6, b=10, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -72 = 172.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √172) / 12.
5. Calculate roots: x1 = 0.25957308738366675, x2 = -1.9262397540503333.
Fi... |
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... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 3, 6, 7, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 3 + 6 + 7 + 2) / 5 = 30 / 5 = 6.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.0, -3.0, 0.0, 1.0, -4.0]
Squared Differences: [36.0, 9.0, 0.0, 1.0, 16.0]
Sum of ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 1, 9, 9, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 1 + 9 + 9 + 2) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, -5.2, 2.8, 2.8, -4.2]
Squared Differences: [14.44, 27.04, 7.84, 7.84, 17.64]
S... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
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) |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-3, 1]
B =
[5, -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) = (2*5) + (3*-2) = 4
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (3*-2) = -12
C21 = (A21 * B11) + (A22 * B21) = (-3*5) + (1*-2) = -17
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (1*-2) = 7
Resul... |
Find the roots of the quadratic equation: 4x^2 + -9x + 8 = 0 | 1. Identify coefficients: a=4, b=-9, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(4)(8) = 81 - 128 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √47i) / 8.
Final Answer: x = 1.125 ± 0.8569568250501305i |
Compute the quotient and remainder of 81343 divided by 42. Show the long division steps. | To divide 81343 by 42:
1. Determine how many times 42 goes into the leading digits of 81343.
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: 81343 / 42 = 1936 with a remainder of 31.
Proof: (1936 * 42... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[3, -3]
B =
[2, 0]
[-1, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*2) + (-4*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (0*0) + (-4*-3) = 12
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-3*-1) = 9
C22 = (A21 * B12) + (A22 * B22) = (3*0) + (-3*-3) = 9
Result M... |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + C |
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 |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Add the fractions 11/2 and 3/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 9, which is 18.
2. Convert 11/2 by multiplying top and bottom by 9: 99/18.
3. Convert 3/9 by multiplying top and bottom by 2: 6/18.
4. Add the numerators: 99 + 6 = 105.
5. The resulting fraction is 105/18.
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) |
Compute the quotient and remainder of 94292 divided by 18. Show the long division steps. | To divide 94292 by 18:
1. Determine how many times 18 goes into the leading digits of 94292.
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: 94292 / 18 = 5238 with a remainder of 8.
Proof: (5238 * 18)... |
Find the roots of the quadratic equation: 3x^2 + -9x + 0 = 0 | 1. Identify coefficients: a=3, b=-9, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 0 = 81.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √81) / 6.
5. Calculate roots: x1 = 3.0, x2 = 0.0.
Final Answer: x = 3.0, 0.0 |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 3^2 = c^2
2. Calculate the squares: 169 + 9 = c^2
3. Sum the squares: 178 = c^2
4. Take the square root of both sides: c = √178
Final Answ... |
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 ... |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 3^2 + 12^2 = c^2
2. Calculate the squares: 9 + 144 = c^2
3. Sum the squares: 153 = c^2
4. Take the square root of both sides: c = √153
Final Answ... |
Multiply the following 2x2 matrices:
A =
[-2, 1]
[4, -2]
B =
[3, 5]
[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) = (-2*3) + (1*0) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*5) + (1*-3) = -13
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (-2*0) = 12
C22 = (A21 * B12) + (A22 * B22) = (4*5) + (-2*-3) = 26
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 3, 6, 11, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 3 + 6 + 11 + 4) / 5 = 29 / 5 = 5.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.8, -2.8, 0.2, 5.2, -1.8]
Squared Differences: [0.64, 7.84, 0.04, 27.04, 3.24]
Su... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 16, 14, 9, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 16 + 14 + 9 + 19) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, 2.6, 0.6, -4.4, 5.6]
Squared Differences: [19.36, 6.76, 0.36, 19.36, 31.36]
... |
Find the roots of the quadratic equation: 6x^2 + 9x + 3 = 0 | 1. Identify coefficients: a=6, b=9, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 72 = 9.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √9) / 12.
5. Calculate roots: x1 = -0.5, x2 = -1.0.
Final Answer: x = -0.5, -1.0 |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Evaluate 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) |
Compute the quotient and remainder of 73660 divided by 39. Show the long division steps. | To divide 73660 by 39:
1. Determine how many times 39 goes into the leading digits of 73660.
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: 73660 / 39 = 1888 with a remainder of 28.
Proof: (1888 * 39... |
Find the roots of the quadratic equation: 2x^2 + -7x + 3 = 0 | 1. Identify coefficients: a=2, b=-7, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 24 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √25) / 4.
5. Calculate roots: x1 = 3.0, x2 = 0.5.
Final Answer: x = 3.0, 0.5 |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^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: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Find the roots of the quadratic equation: 1x^2 + 1x + -5 = 0 | 1. Identify coefficients: a=1, b=1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -20 = 21.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √21) / 2.
5. Calculate roots: x1 = 1.79128784747792, x2 = -2.79128784747792.
Final Answer: ... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 6, 10, 7, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 6 + 10 + 7 + 13) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, -3.2, 0.8, -2.2, 3.8]
Squared Differences: [0.64, 10.24, 0.64, 4.84, 14.44]
... |
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 ... |
Add the fractions 5/3 and 12/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 14, which is 42.
2. Convert 5/3 by multiplying top and bottom by 14: 70/42.
3. Convert 12/14 by multiplying top and bottom by 3: 36/42.
4. Add the numerators: 70 + 36 = 106.
5. The resulting fraction is 106/42.
6. Simplify ... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 9^2 = c^2
2. Calculate the squares: 225 + 81 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
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 =
[1, -3]
[-3, 5]
B =
[3, -1]
[2, -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*3) + (-3*2) = -3
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (-3*-3) = 8
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (5*2) = 1
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (5*-3) = -12
Result... |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 4^2 + 12^2 = c^2
2. Calculate the squares: 16 + 144 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
Final Ans... |
Compute the quotient and remainder of 11444 divided by 32. Show the long division steps. | To divide 11444 by 32:
1. Determine how many times 32 goes into the leading digits of 11444.
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: 11444 / 32 = 357 with a remainder of 20.
Proof: (357 * 32) ... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
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 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 |
Compute the indefinite integral of f(x) = 3x^3. | To find the indefinite integral ∫ 3x^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: 3 / 4 = 3/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^3 dx = 3/4x^4 + C |
Add the fractions 3/7 and 15/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 5, which is 35.
2. Convert 3/7 by multiplying top and bottom by 5: 15/35.
3. Convert 15/5 by multiplying top and bottom by 7: 105/35.
4. Add the numerators: 15 + 105 = 120.
5. The resulting fraction is 120/35.
6. Simplify b... |
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 roots of the quadratic equation: 3x^2 + -8x + -10 = 0 | 1. Identify coefficients: a=3, b=-8, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -120 = 184.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √184) / 6.
5. Calculate roots: x1 = 3.59410999437509, x2 = -0.9274433277084227.
Final A... |
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_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 91597 divided by 84. Show the long division steps. | To divide 91597 by 84:
1. Determine how many times 84 goes into the leading digits of 91597.
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: 91597 / 84 = 1090 with a remainder of 37.
Proof: (1090 * 84... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + 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 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 ... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Find the derivative of f(x) = (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 73687 divided by 56. Show the long division steps. | To divide 73687 by 56:
1. Determine how many times 56 goes into the leading digits of 73687.
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: 73687 / 56 = 1315 with a remainder of 47.
Proof: (1315 * 56... |
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