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Find the roots of the quadratic equation: 3x^2 + 3x + 3 = 0 | 1. Identify coefficients: a=3, b=3, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(3)(3) = 9 - 36 = -27.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √27i) / 6.
Final Answer: x = -0.5 ± 0.8660254037844387i |
Compute the quotient and remainder of 66526 divided by 46. Show the long division steps. | To divide 66526 by 46:
1. Determine how many times 46 goes into the leading digits of 66526.
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: 66526 / 46 = 1446 with a remainder of 10.
Proof: (1446 * 46... |
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 +... |
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 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) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 5, 8, 1, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 5 + 8 + 1 + 3) / 5 = 32 / 5 = 6.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.6, -1.4, 1.6, -5.4, -3.4]
Squared Differences: [73.96, 1.96, 2.56, 29.16, 11.56]
... |
Compute the quotient and remainder of 33431 divided by 46. Show the long division steps. | To divide 33431 by 46:
1. Determine how many times 46 goes into the leading digits of 33431.
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: 33431 / 46 = 726 with a remainder of 35.
Proof: (726 * 46) ... |
Add the fractions 5/8 and 1/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 7, which is 56.
2. Convert 5/8 by multiplying top and bottom by 7: 35/56.
3. Convert 1/7 by multiplying top and bottom by 8: 8/56.
4. Add the numerators: 35 + 8 = 43.
5. The resulting fraction is 43/56.
Final Answer: 43/56 |
Find the roots of the quadratic equation: 10x^2 + -7x + 2 = 0 | 1. Identify coefficients: a=10, b=-7, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(10)(2) = 49 - 80 = -31.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √31i) / 20.
Final Answer: x = 0.35 ± 0.27838821814150105i |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + -5x + 2 = 0 | 1. Identify coefficients: a=8, b=-5, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(8)(2) = 25 - 64 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √39i) / 16.
Final Answer: x = 0.3125 ± 0.3903123748998999i |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[5, 2]
B =
[3, -4]
[4, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*3) + (4*4) = 7
C12 = (A11 * B12) + (A12 * B22) = (-3*-4) + (4*-4) = -4
C21 = (A21 * B11) + (A22 * B21) = (5*3) + (2*4) = 23
C22 = (A21 * B12) + (A22 * B22) = (5*-4) + (2*-4) = -28
Result ... |
Compute the quotient and remainder of 70594 divided by 10. Show the long division steps. | To divide 70594 by 10:
1. Determine how many times 10 goes into the leading digits of 70594.
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: 70594 / 10 = 7059 with a remainder of 4.
Proof: (7059 * 10)... |
Find the roots of the quadratic equation: 10x^2 + 10x + -2 = 0 | 1. Identify coefficients: a=10, b=10, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -80 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √180) / 20.
5. Calculate roots: x1 = 0.17082039324993695, x2 = -1.170820393249937.
Fi... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 13^2 = c^2
2. Calculate the squares: 400 + 169 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
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 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 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 |
Multiply the following 2x2 matrices:
A =
[2, 5]
[-1, 0]
B =
[-2, -2]
[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) = (2*-2) + (5*3) = 11
C12 = (A11 * B12) + (A12 * B22) = (2*-2) + (5*-4) = -24
C21 = (A21 * B11) + (A22 * B21) = (-1*-2) + (0*3) = 2
C22 = (A21 * B12) + (A22 * B22) = (-1*-2) + (0*-4) = 2
Result... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 10^2 = c^2
2. Calculate the squares: 225 + 100 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final A... |
Add the fractions 15/9 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 5, which is 45.
2. Convert 15/9 by multiplying top and bottom by 5: 75/45.
3. Convert 4/5 by multiplying top and bottom by 9: 36/45.
4. Add the numerators: 75 + 36 = 111.
5. The resulting fraction is 111/45.
6. Simplify by ... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 14, 6, 9, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 14 + 6 + 9 + 20) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, 1.4, -6.6, -3.6, 7.4]
Squared Differences: [1.96, 1.96, 43.56, 12.96, 54.76]
... |
Multiply the following 2x2 matrices:
A =
[3, -2]
[-5, -3]
B =
[0, 0]
[-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) = (3*0) + (-2*-3) = 6
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (-2*2) = -4
C21 = (A21 * B11) + (A22 * B21) = (-5*0) + (-3*-3) = 9
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (-3*2) = -6
Result ... |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[-2, -2]
B =
[0, 5]
[-2, -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) = (-2*0) + (2*-2) = -4
C12 = (A11 * B12) + (A12 * B22) = (-2*5) + (2*-4) = -18
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-2*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (-2*-4) = -2
Res... |
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 |
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 = 11 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: 11^2 + 3^2 = c^2
2. Calculate the squares: 121 + 9 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answ... |
Multiply the following 2x2 matrices:
A =
[3, 2]
[-3, 3]
B =
[3, 0]
[-2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*3) + (2*-2) = 5
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (2*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (3*-2) = -15
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (3*0) = 0
Result Matr... |
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 ... |
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) |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[1, 1]
B =
[0, 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) = (-3*0) + (4*-1) = -4
C12 = (A11 * B12) + (A12 * B22) = (-3*0) + (4*2) = 8
C21 = (A21 * B11) + (A22 * B21) = (1*0) + (1*-1) = -1
C22 = (A21 * B12) + (A22 * B22) = (1*0) + (1*2) = 2
Result Matr... |
Multiply the following 2x2 matrices:
A =
[1, 2]
[2, -2]
B =
[-5, 5]
[-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) = (1*-5) + (2*-1) = -7
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (2*3) = 11
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-2*-1) = -8
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (-2*3) = 4
Result M... |
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 |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Add the fractions 10/7 and 11/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 12, which is 84.
2. Convert 10/7 by multiplying top and bottom by 12: 120/84.
3. Convert 11/12 by multiplying top and bottom by 7: 77/84.
4. Add the numerators: 120 + 77 = 197.
5. The resulting fraction is 197/84.
Final Ans... |
Compute the quotient and remainder of 51278 divided by 49. Show the long division steps. | To divide 51278 by 49:
1. Determine how many times 49 goes into the leading digits of 51278.
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: 51278 / 49 = 1046 with a remainder of 24.
Proof: (1046 * 49... |
Find the roots of the quadratic equation: 2x^2 + -8x + -6 = 0 | 1. Identify coefficients: a=2, b=-8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -48 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √112) / 4.
5. Calculate roots: x1 = 4.645751311064591, x2 = -0.6457513110645907.
Final An... |
Multiply the following 2x2 matrices:
A =
[2, 2]
[1, 3]
B =
[1, 5]
[5, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*1) + (2*5) = 12
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (2*-5) = 0
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (3*5) = 16
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (3*-5) = -10
Result Matr... |
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 +... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[1, 4]
B =
[-4, -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) = (2*-4) + (3*-4) = -20
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (3*-5) = -17
C21 = (A21 * B11) + (A22 * B21) = (1*-4) + (4*-4) = -20
C22 = (A21 * B12) + (A22 * B22) = (1*-1) + (4*-5) = -21
R... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 12^2 = c^2
2. Calculate the squares: 64 + 144 = c^2
3. Sum the squares: 208 = c^2
4. Take the square root of both sides: c = √208
Final Ans... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 2x^2 + 9x + 4 = 0 | 1. Identify coefficients: a=2, b=9, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 32 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √49) / 4.
5. Calculate roots: x1 = -0.5, x2 = -4.0.
Final Answer: x = -0.5, -4.0 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 12^2 = c^2
2. Calculate the squares: 196 + 144 = c^2
3. Sum the squares: 340 = c^2
4. Take the square root of both sides: c = √340
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 8^2 = c^2
2. Calculate the squares: 25 + 64 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 19, 17, 19, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 19 + 17 + 19 + 5) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.6, 4.4, 2.4, 4.4, -9.6]
Squared Differences: [2.56, 19.36, 5.76, 19.36, 92.16]
... |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[5, 0]
B =
[-2, 4]
[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) = (-2*-2) + (-5*1) = -1
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (-5*0) = -8
C21 = (A21 * B11) + (A22 * B21) = (5*-2) + (0*1) = -10
C22 = (A21 * B12) + (A22 * B22) = (5*4) + (0*0) = 20
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 8, 7, 19, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 8 + 7 + 19 + 9) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.6, -4.4, -5.4, 6.6, -3.4]
Squared Differences: [43.56, 19.36, 29.16, 43.56, 11.56]... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, 0]
[2, 2]
B =
[-4, -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) = (-3*-4) + (0*-5) = 12
C12 = (A11 * B12) + (A12 * B22) = (-3*-4) + (0*0) = 12
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (2*-5) = -18
C22 = (A21 * B12) + (A22 * B22) = (2*-4) + (2*0) = -8
Resu... |
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) |
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 |
Find the roots of the quadratic equation: 8x^2 + -3x + -2 = 0 | 1. Identify coefficients: a=8, b=-3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -64 = 73.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √73) / 16.
5. Calculate roots: x1 = 0.7215002340823456, x2 = -0.34650023408234565.
Final An... |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + -9x + -6 = 0 | 1. Identify coefficients: a=8, b=-9, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -192 = 273.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √273) / 16.
5. Calculate roots: x1 = 1.595169477616144, x2 = -0.470169477616144.
Final A... |
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) |
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 +... |
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) = 0
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-1*-2) = 17
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (-1*-2) = 11
... |
Find the roots of the quadratic equation: 5x^2 + 0x + -10 = 0 | 1. Identify coefficients: a=5, b=0, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -200 = 200.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √200) / 10.
5. Calculate roots: x1 = 1.4142135623730951, x2 = -1.4142135623730951.
Final ... |
Add the fractions 10/14 and 14/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 13, which is 182.
2. Convert 10/14 by multiplying top and bottom by 13: 130/182.
3. Convert 14/13 by multiplying top and bottom by 14: 196/182.
4. Add the numerators: 130 + 196 = 326.
5. The resulting fraction is 326/182.
... |
Compute the quotient and remainder of 15176 divided by 54. Show the long division steps. | To divide 15176 by 54:
1. Determine how many times 54 goes into the leading digits of 15176.
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: 15176 / 54 = 281 with a remainder of 2.
Proof: (281 * 54) +... |
Find the roots of the quadratic equation: 5x^2 + 6x + -4 = 0 | 1. Identify coefficients: a=5, b=6, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -80 = 116.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √116) / 10.
5. Calculate roots: x1 = 0.47703296142690077, x2 = -1.6770329614269008.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 13^2 = c^2
2. Calculate the squares: 144 + 169 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
Find the roots of the quadratic equation: 9x^2 + 5x + 7 = 0 | 1. Identify coefficients: a=9, b=5, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(9)(7) = 25 - 252 = -227.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √227i) / 18.
Final Answer: x = -0.2777777777777778 ± 0.8370288429621868i |
Compute the quotient and remainder of 97763 divided by 44. Show the long division steps. | To divide 97763 by 44:
1. Determine how many times 44 goes into the leading digits of 97763.
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: 97763 / 44 = 2221 with a remainder of 39.
Proof: (2221 * 44... |
Find the roots of the quadratic equation: 6x^2 + -4x + 6 = 0 | 1. Identify coefficients: a=6, b=-4, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(6)(6) = 16 - 144 = -128.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √128i) / 12.
Final Answer: x = 0.3333333333333333 ± 0.9428090415820635i |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 10^2 = c^2
2. Calculate the squares: 324 + 100 = c^2
3. Sum the squares: 424 = c^2
4. Take the square root of both sides: c = √424
Final A... |
Multiply the following 2x2 matrices:
A =
[4, -3]
[-5, -2]
B =
[0, 2]
[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*0) + (-3*5) = -15
C12 = (A11 * B12) + (A12 * B22) = (4*2) + (-3*0) = 8
C21 = (A21 * B11) + (A22 * B21) = (-5*0) + (-2*5) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (-2*0) = -10
Resul... |
Add the fractions 1/9 and 2/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 9, which is 9.
2. Convert 1/9 by multiplying top and bottom by 1: 1/9.
3. Convert 2/9 by multiplying top and bottom by 1: 2/9.
4. Add the numerators: 1 + 2 = 3.
5. The resulting fraction is 3/9.
6. Simplify by dividing nume... |
Add the fractions 13/8 and 4/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 3, which is 24.
2. Convert 13/8 by multiplying top and bottom by 3: 39/24.
3. Convert 4/3 by multiplying top and bottom by 8: 32/24.
4. Add the numerators: 39 + 32 = 71.
5. The resulting fraction is 71/24.
Final Answer: 71/... |
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 9/9 and 13/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 4, which is 36.
2. Convert 9/9 by multiplying top and bottom by 4: 36/36.
3. Convert 13/4 by multiplying top and bottom by 9: 117/36.
4. Add the numerators: 36 + 117 = 153.
5. The resulting fraction is 153/36.
6. Simplify b... |
Find the roots of the quadratic equation: 4x^2 + -5x + -6 = 0 | 1. Identify coefficients: a=4, b=-5, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -96 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √121) / 8.
5. Calculate roots: x1 = 2.0, x2 = -0.75.
Final Answer: x = 2.0, -0.75 |
Add the fractions 11/15 and 11/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 4, which is 60.
2. Convert 11/15 by multiplying top and bottom by 4: 44/60.
3. Convert 11/4 by multiplying top and bottom by 15: 165/60.
4. Add the numerators: 44 + 165 = 209.
5. The resulting fraction is 209/60.
Final Ans... |
Add the fractions 15/14 and 4/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 11, which is 154.
2. Convert 15/14 by multiplying top and bottom by 11: 165/154.
3. Convert 4/11 by multiplying top and bottom by 14: 56/154.
4. Add the numerators: 165 + 56 = 221.
5. The resulting fraction is 221/154.
Fin... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 11^2 = c^2
2. Calculate the squares: 81 + 121 = c^2
3. Sum the squares: 202 = c^2
4. Take the square root of both sides: c = √202
Final Ans... |
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 |
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... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 19, 9, 3, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 19 + 9 + 3 + 4) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, 9.0, -1.0, -7.0, -6.0]
Squared Differences: [25.0, 81.0, 1.0, 49.0, 36.0]
Su... |
Add the fractions 14/14 and 9/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 7, which is 14.
2. Convert 14/14 by multiplying top and bottom by 1: 14/14.
3. Convert 9/7 by multiplying top and bottom by 2: 18/14.
4. Add the numerators: 14 + 18 = 32.
5. The resulting fraction is 32/14.
6. Simplify by ... |
Add the fractions 5/9 and 9/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 10, which is 90.
2. Convert 5/9 by multiplying top and bottom by 10: 50/90.
3. Convert 9/10 by multiplying top and bottom by 9: 81/90.
4. Add the numerators: 50 + 81 = 131.
5. The resulting fraction is 131/90.
Final Answer:... |
Find the roots of the quadratic equation: 1x^2 + 4x + -3 = 0 | 1. Identify coefficients: a=1, b=4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -12 = 28.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √28) / 2.
5. Calculate roots: x1 = 0.6457513110645907, x2 = -4.645751311064591.
Final Answ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 7^2 = c^2
2. Calculate the squares: 169 + 49 = c^2
3. Sum the squares: 218 = c^2
4. Take the square root of both sides: c = √218
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 |
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 |
Add the fractions 12/11 and 3/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 2, which is 22.
2. Convert 12/11 by multiplying top and bottom by 2: 24/22.
3. Convert 3/2 by multiplying top and bottom by 11: 33/22.
4. Add the numerators: 24 + 33 = 57.
5. The resulting fraction is 57/22.
Final Answer: ... |
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 |
Multiply the following 2x2 matrices:
A =
[3, 4]
[-1, 1]
B =
[5, -2]
[-4, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*5) + (4*-4) = -1
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (4*3) = 6
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (1*-4) = -9
C22 = (A21 * B12) + (A22 * B22) = (-1*-2) + (1*3) = 5
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 14. 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 + 14^2 = c^2
2. Calculate the squares: 225 + 196 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 11, 17, 18, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 11 + 17 + 18 + 20) / 5 = 83 / 5 = 16.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, -5.6, 0.4, 1.4, 3.4]
Squared Differences: [0.16, 31.36, 0.16, 1.96, 11.56]
... |
Multiply the following 2x2 matrices:
A =
[2, 0]
[1, 1]
B =
[5, -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) = (2*5) + (0*1) = 10
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (0*5) = -2
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (1*1) = 6
C22 = (A21 * B12) + (A22 * B22) = (1*-1) + (1*5) = 4
Result Matrix... |
Compute the quotient and remainder of 90861 divided by 50. Show the long division steps. | To divide 90861 by 50:
1. Determine how many times 50 goes into the leading digits of 90861.
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: 90861 / 50 = 1817 with a remainder of 11.
Proof: (1817 * 50... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 6^2 = c^2
2. Calculate the squares: 25 + 36 = c^2
3. Sum the squares: 61 = c^2
4. Take the square root of both sides: c = √61
Final Answer:... |
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) |
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