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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 ... |
Compute the quotient and remainder of 56707 divided by 30. Show the long division steps. | To divide 56707 by 30:
1. Determine how many times 30 goes into the leading digits of 56707.
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: 56707 / 30 = 1890 with a remainder of 7.
Proof: (1890 * 30)... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^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: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
Compute the quotient and remainder of 64273 divided by 72. Show the long division steps. | To divide 64273 by 72:
1. Determine how many times 72 goes into the leading digits of 64273.
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: 64273 / 72 = 892 with a remainder of 49.
Proof: (892 * 72) ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 9^2 = c^2
2. Calculate the squares: 256 + 81 = c^2
3. Sum the squares: 337 = c^2
4. Take the square root of both sides: c = √337
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-4, 5]
[-5, 1]
B =
[-5, 4]
[-5, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (5*-5) = -5
C12 = (A11 * B12) + (A12 * B22) = (-4*4) + (5*0) = -16
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (1*-5) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (1*0) = -20
Res... |
Compute the quotient and remainder of 45282 divided by 71. Show the long division steps. | To divide 45282 by 71:
1. Determine how many times 71 goes into the leading digits of 45282.
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: 45282 / 71 = 637 with a remainder of 55.
Proof: (637 * 71) ... |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) 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) |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 7^2 = c^2
2. Calculate the squares: 324 + 49 = c^2
3. Sum the squares: 373 = c^2
4. Take the square root of both sides: c = √373
Final Ans... |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
Find the roots of the quadratic equation: 5x^2 + 3x + 1 = 0 | 1. Identify coefficients: a=5, b=3, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(5)(1) = 9 - 20 = -11.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √11i) / 10.
Final Answer: x = -0.3 ± 0.33166247903553997i |
Compute the quotient and remainder of 58055 divided by 10. Show the long division steps. | To divide 58055 by 10:
1. Determine how many times 10 goes into the leading digits of 58055.
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: 58055 / 10 = 5805 with a remainder of 5.
Proof: (5805 * 10)... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 14, 11, 7, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 14 + 11 + 7 + 10) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 4.2, 1.2, -2.8, 0.2]
Squared Differences: [7.84, 17.64, 1.44, 7.84, 0.04]
S... |
Compute the quotient and remainder of 5862 divided by 91. Show the long division steps. | To divide 5862 by 91:
1. Determine how many times 91 goes into the leading digits of 5862.
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: 5862 / 91 = 64 with a remainder of 38.
Proof: (64 * 91) + 38 ... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^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: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
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 |
Find the roots of the quadratic equation: 6x^2 + -10x + 2 = 0 | 1. Identify coefficients: a=6, b=-10, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 48 = 52.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √52) / 12.
5. Calculate roots: x1 = 1.434258545910665, x2 = 0.2324081207560018.
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 3^2 = c^2
2. Calculate the squares: 289 + 9 = c^2
3. Sum the squares: 298 = c^2
4. Take the square root of both sides: c = √298
Final Answ... |
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) |
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 sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the 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 = 12 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: 12^2 + 4^2 = c^2
2. Calculate the squares: 144 + 16 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
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) |
Add the fractions 5/4 and 1/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 6, which is 12.
2. Convert 5/4 by multiplying top and bottom by 3: 15/12.
3. Convert 1/6 by multiplying top and bottom by 2: 2/12.
4. Add the numerators: 15 + 2 = 17.
5. The resulting fraction is 17/12.
Final Answer: 17/12 |
Compute the quotient and remainder of 11281 divided by 42. Show the long division steps. | To divide 11281 by 42:
1. Determine how many times 42 goes into the leading digits of 11281.
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: 11281 / 42 = 268 with a remainder of 25.
Proof: (268 * 42) ... |
Compute the quotient and remainder of 14438 divided by 13. Show the long division steps. | To divide 14438 by 13:
1. Determine how many times 13 goes into the leading digits of 14438.
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: 14438 / 13 = 1110 with a remainder of 8.
Proof: (1110 * 13)... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[5, -2]
B =
[4, -1]
[-1, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*4) + (3*-1) = -7
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (3*-5) = -14
C21 = (A21 * B11) + (A22 * B21) = (5*4) + (-2*-1) = 22
C22 = (A21 * B12) + (A22 * B22) = (5*-1) + (-2*-5) = 5
Res... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 18^2 = c^2
2. Calculate the squares: 36 + 324 = c^2
3. Sum the squares: 360 = c^2
4. Take the square root of both sides: c = √360
Final Ans... |
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 = 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... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[0, 1]
B =
[-5, 3]
[-5, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (-1*-5) = 25
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (-1*3) = -15
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (1*-5) = -5
C22 = (A21 * B12) + (A22 * B22) = (0*3) + (1*3) = 3
Resul... |
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 ... |
Compute the indefinite integral of f(x) = 10x^2. | To find the indefinite integral ∫ 10x^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: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 2, 13, 17, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 2 + 13 + 17 + 11) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.0, -9.0, 2.0, 6.0, 0.0]
Squared Differences: [1.0, 81.0, 4.0, 36.0, 0.0]
Sum ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 17^2 = c^2
2. Calculate the squares: 400 + 289 = c^2
3. Sum the squares: 689 = c^2
4. Take the square root of both sides: c = √689
Final A... |
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 |
Compute the quotient and remainder of 63899 divided by 56. Show the long division steps. | To divide 63899 by 56:
1. Determine how many times 56 goes into the leading digits of 63899.
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: 63899 / 56 = 1141 with a remainder of 3.
Proof: (1141 * 56)... |
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) |
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 |
Add the fractions 12/4 and 5/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 15, which is 60.
2. Convert 12/4 by multiplying top and bottom by 15: 180/60.
3. Convert 5/15 by multiplying top and bottom by 4: 20/60.
4. Add the numerators: 180 + 20 = 200.
5. The resulting fraction is 200/60.
6. Simplif... |
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 77214 divided by 25. Show the long division steps. | To divide 77214 by 25:
1. Determine how many times 25 goes into the leading digits of 77214.
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: 77214 / 25 = 3088 with a remainder of 14.
Proof: (3088 * 25... |
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 |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) 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) |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[4, 4]
B =
[-2, 5]
[4, 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*-2) + (4*4) = 20
C12 = (A11 * B12) + (A12 * B22) = (-2*5) + (4*1) = -6
C21 = (A21 * B11) + (A22 * B21) = (4*-2) + (4*4) = 8
C22 = (A21 * B12) + (A22 * B22) = (4*5) + (4*1) = 24
Result Mat... |
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 |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[4, 1]
B =
[-5, 1]
[0, -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*-5) + (-5*0) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*1) + (-5*-1) = 3
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (1*0) = -20
C22 = (A21 * B12) + (A22 * B22) = (4*1) + (1*-1) = 3
Result... |
Find the roots of the quadratic equation: 2x^2 + -3x + 0 = 0 | 1. Identify coefficients: a=2, b=-3, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - 0 = 9.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √9) / 4.
5. Calculate roots: x1 = 1.5, x2 = 0.0.
Final Answer: x = 1.5, 0.0 |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 20, 11, 13, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 20 + 11 + 13 + 7) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 6.6, -2.4, -0.4, -6.4]
Squared Differences: [6.76, 43.56, 5.76, 0.16, 40.96]
... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 9, 17, 2, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 9 + 17 + 2 + 6) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, 0.6, 8.6, -6.4, -2.4]
Squared Differences: [0.16, 0.36, 73.96, 40.96, 5.76]
S... |
Find the roots of the quadratic equation: 8x^2 + 4x + 0 = 0 | 1. Identify coefficients: a=8, b=4, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - 0 = 16.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √16) / 16.
5. Calculate roots: x1 = 0.0, x2 = -0.5.
Final Answer: x = 0.0, -0.5 |
Add the fractions 15/11 and 14/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 9, which is 99.
2. Convert 15/11 by multiplying top and bottom by 9: 135/99.
3. Convert 14/9 by multiplying top and bottom by 11: 154/99.
4. Add the numerators: 135 + 154 = 289.
5. The resulting fraction is 289/99.
Final A... |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 1x^2 + 5x + 6 = 0 | 1. Identify coefficients: a=1, b=5, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 24 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √1) / 2.
5. Calculate roots: x1 = -2.0, x2 = -3.0.
Final Answer: x = -2.0, -3.0 |
Add the fractions 9/15 and 4/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 7, which is 105.
2. Convert 9/15 by multiplying top and bottom by 7: 63/105.
3. Convert 4/7 by multiplying top and bottom by 15: 60/105.
4. Add the numerators: 63 + 60 = 123.
5. The resulting fraction is 123/105.
6. Simpli... |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^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: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 18, 12, 19, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 18 + 12 + 19 + 17) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, 3.4, -2.6, 4.4, 2.4]
Squared Differences: [57.76, 11.56, 6.76, 19.36, 5.76]
... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[3, -3]
B =
[-4, -3]
[-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) = (4*-4) + (0*-1) = -16
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (0*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (3*-4) + (-3*-1) = -9
C22 = (A21 * B12) + (A22 * B22) = (3*-3) + (-3*0) = -9
Res... |
Add the fractions 14/3 and 6/2. 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 14/3 by multiplying top and bottom by 2: 28/6.
3. Convert 6/2 by multiplying top and bottom by 3: 18/6.
4. Add the numerators: 28 + 18 = 46.
5. The resulting fraction is 46/6.
6. Simplify by dividi... |
Compute the quotient and remainder of 80334 divided by 22. Show the long division steps. | To divide 80334 by 22:
1. Determine how many times 22 goes into the leading digits of 80334.
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: 80334 / 22 = 3651 with a remainder of 12.
Proof: (3651 * 22... |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 98544 divided by 63. Show the long division steps. | To divide 98544 by 63:
1. Determine how many times 63 goes into the leading digits of 98544.
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: 98544 / 63 = 1564 with a remainder of 12.
Proof: (1564 * 63... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 3^2 = c^2
2. Calculate the squares: 49 + 9 = c^2
3. Sum the squares: 58 = c^2
4. Take the square root of both sides: c = √58
Final Answer: ... |
Add the fractions 15/8 and 4/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 15/8 by multiplying top and bottom by 1: 15/8.
3. Convert 4/8 by multiplying top and bottom by 1: 4/8.
4. Add the numerators: 15 + 4 = 19.
5. The resulting fraction is 19/8.
Final Answer: 19/8 |
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 = 8^2 - 4(10)(7) = 64 - 280 = -216.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √216i) / 20.
Final Answer: x = -0.4 ± 0.7348469228349535i |
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 15/12. 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 11/2 by multiplying top and bottom by 6: 66/12.
3. Convert 15/12 by multiplying top and bottom by 1: 15/12.
4. Add the numerators: 66 + 15 = 81.
5. The resulting fraction is 81/12.
6. Simplify by... |
Find the roots of the quadratic equation: 2x^2 + 10x + 1 = 0 | 1. Identify coefficients: a=2, b=10, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 8 = 92.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √92) / 4.
5. Calculate roots: x1 = -0.10208423834364044, x2 = -4.89791576165636.
Final Ans... |
Compute the quotient and remainder of 81505 divided by 39. Show the long division steps. | To divide 81505 by 39:
1. Determine how many times 39 goes into the leading digits of 81505.
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: 81505 / 39 = 2089 with a remainder of 34.
Proof: (2089 * 39... |
Compute the indefinite integral of f(x) = 4x^3. | To find the indefinite integral ∫ 4x^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: 4 / 4 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^3 dx = 1x^4 + C |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[0, -3]
[-5, -4]
B =
[2, -5]
[0, 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) + (-3*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*-5) + (-3*1) = -3
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (-4*0) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-4*1) = 21
Resul... |
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_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 11/9 and 4/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 8, which is 72.
2. Convert 11/9 by multiplying top and bottom by 8: 88/72.
3. Convert 4/8 by multiplying top and bottom by 9: 36/72.
4. Add the numerators: 88 + 36 = 124.
5. The resulting fraction is 124/72.
6. Simplify by ... |
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 ... |
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 98388 divided by 63. Show the long division steps. | To divide 98388 by 63:
1. Determine how many times 63 goes into the leading digits of 98388.
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: 98388 / 63 = 1561 with a remainder of 45.
Proof: (1561 * 63... |
Find the roots of the quadratic equation: 9x^2 + -1x + -6 = 0 | 1. Identify coefficients: a=9, b=-1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -216 = 217.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √217) / 18.
5. Calculate roots: x1 = 0.8739399923697908, x2 = -0.7628288812586796.
Final ... |
Multiply the following 2x2 matrices:
A =
[5, -4]
[-4, -1]
B =
[-1, -2]
[-3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-1) + (-4*-3) = 7
C12 = (A11 * B12) + (A12 * B22) = (5*-2) + (-4*-2) = -2
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (-1*-3) = 7
C22 = (A21 * B12) + (A22 * B22) = (-4*-2) + (-1*-2) = 10
R... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 7, 17, 18, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 7 + 17 + 18 + 18) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, -6.8, 3.2, 4.2, 4.2]
Squared Differences: [23.04, 46.24, 10.24, 17.64, 17.64]... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 13^2 = c^2
2. Calculate the squares: 81 + 169 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
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 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... |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
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 |
Find the roots of the quadratic equation: 5x^2 + -4x + -3 = 0 | 1. Identify coefficients: a=5, b=-4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -60 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √76) / 10.
5. Calculate roots: x1 = 1.2717797887081348, x2 = -0.4717797887081348.
Final An... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 12^2 = c^2
2. Calculate the squares: 400 + 144 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Compute the quotient and remainder of 57595 divided by 38. Show the long division steps. | To divide 57595 by 38:
1. Determine how many times 38 goes into the leading digits of 57595.
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: 57595 / 38 = 1515 with a remainder of 25.
Proof: (1515 * 38... |
Multiply the following 2x2 matrices:
A =
[1, 4]
[0, 3]
B =
[2, -5]
[0, 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*2) + (4*0) = 2
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (4*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (0*2) + (3*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (3*1) = 3
Result Matrix ... |
Compute the quotient and remainder of 80934 divided by 89. Show the long division steps. | To divide 80934 by 89:
1. Determine how many times 89 goes into the leading digits of 80934.
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: 80934 / 89 = 909 with a remainder of 33.
Proof: (909 * 89) ... |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[-1, -3]
[5, 3]
B =
[0, 1]
[-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*0) + (-3*-4) = 12
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (-3*5) = -16
C21 = (A21 * B11) + (A22 * B21) = (5*0) + (3*-4) = -12
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (3*5) = 20
Resul... |
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