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Find the roots of the quadratic equation: 4x^2 + -3x + 8 = 0 | 1. Identify coefficients: a=4, b=-3, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(4)(8) = 9 - 128 = -119.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √119i) / 8.
Final Answer: x = 0.375 ± 1.3635890143294642i |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 9^2 = c^2
2. Calculate the squares: 16 + 81 = c^2
3. Sum the squares: 97 = c^2
4. Take the square root of both sides: c = √97
Final Answer:... |
Compute the quotient and remainder of 49544 divided by 19. Show the long division steps. | To divide 49544 by 19:
1. Determine how many times 19 goes into the leading digits of 49544.
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: 49544 / 19 = 2607 with a remainder of 11.
Proof: (2607 * 19... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 10, 13, 7, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 10 + 13 + 7 + 2) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.8, 0.8, 3.8, -2.2, -7.2]
Squared Differences: [23.04, 0.64, 14.44, 4.84, 51.84]
... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 6^2 = c^2
2. Calculate the squares: 289 + 36 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final Ans... |
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 +... |
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... |
Compute the quotient and remainder of 20379 divided by 41. Show the long division steps. | To divide 20379 by 41:
1. Determine how many times 41 goes into the leading digits of 20379.
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: 20379 / 41 = 497 with a remainder of 2.
Proof: (497 * 41) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 13, 18, 14, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 13 + 18 + 14 + 13) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, -1.6, 3.4, -0.6, -1.6]
Squared Differences: [0.16, 2.56, 11.56, 0.36, 2.56]
... |
Find the roots of the quadratic equation: 8x^2 + 9x + -4 = 0 | 1. Identify coefficients: a=8, b=9, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -128 = 209.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √209) / 16.
5. Calculate roots: x1 = 0.34105201842506006, x2 = -1.46605201842506.
Final ... |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^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: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + C |
Compute the quotient and remainder of 7839 divided by 4. Show the long division steps. | To divide 7839 by 4:
1. Determine how many times 4 goes into the leading digits of 7839.
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: 7839 / 4 = 1959 with a remainder of 3.
Proof: (1959 * 4) + 3 = ... |
Compute the quotient and remainder of 76671 divided by 4. Show the long division steps. | To divide 76671 by 4:
1. Determine how many times 4 goes into the leading digits of 76671.
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: 76671 / 4 = 19167 with a remainder of 3.
Proof: (19167 * 4) +... |
Compute the quotient and remainder of 14512 divided by 31. Show the long division steps. | To divide 14512 by 31:
1. Determine how many times 31 goes into the leading digits of 14512.
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: 14512 / 31 = 468 with a remainder of 4.
Proof: (468 * 31) +... |
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 98230 divided by 96. Show the long division steps. | To divide 98230 by 96:
1. Determine how many times 96 goes into the leading digits of 98230.
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: 98230 / 96 = 1023 with a remainder of 22.
Proof: (1023 * 96... |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[0, -2]
B =
[-1, -2]
[-2, 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) + (-5*-2) = 12
C12 = (A11 * B12) + (A12 * B22) = (-2*-2) + (-5*5) = -21
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (-2*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (-2*5) = -10
... |
Add the fractions 8/9 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 8/9 by multiplying top and bottom by 10: 80/90.
3. Convert 9/10 by multiplying top and bottom by 9: 81/90.
4. Add the numerators: 80 + 81 = 161.
5. The resulting fraction is 161/90.
Final Answer:... |
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... |
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) |
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) |
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 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) |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 5, 14, 7, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 5 + 14 + 7 + 12) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.8, -6.2, 2.8, -4.2, 0.8]
Squared Differences: [46.24, 38.44, 7.84, 17.64, 0.64]
... |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Add the fractions 1/4 and 4/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 13, which is 52.
2. Convert 1/4 by multiplying top and bottom by 13: 13/52.
3. Convert 4/13 by multiplying top and bottom by 4: 16/52.
4. Add the numerators: 13 + 16 = 29.
5. The resulting fraction is 29/52.
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 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 cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Add the fractions 4/8 and 1/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 4/8 by multiplying top and bottom by 15: 60/120.
3. Convert 1/15 by multiplying top and bottom by 8: 8/120.
4. Add the numerators: 60 + 8 = 68.
5. The resulting fraction is 68/120.
6. Simplify b... |
Compute the indefinite integral of f(x) = 6x^2. | To find the indefinite integral ∫ 6x^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: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + C |
Find the roots of the quadratic equation: 6x^2 + -3x + -7 = 0 | 1. Identify coefficients: a=6, b=-3, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -168 = 177.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √177) / 12.
5. Calculate roots: x1 = 1.3586778913041726, x2 = -0.8586778913041725.
Final ... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 15^2 = c^2
2. Calculate the squares: 36 + 225 = c^2
3. Sum the squares: 261 = c^2
4. Take the square root of both sides: c = √261
Final Ans... |
Add the fractions 13/7 and 13/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 13, which is 91.
2. Convert 13/7 by multiplying top and bottom by 13: 169/91.
3. Convert 13/13 by multiplying top and bottom by 7: 91/91.
4. Add the numerators: 169 + 91 = 260.
5. The resulting fraction is 260/91.
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 |
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 |
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 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 ... |
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 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √49) / 2.
5. Calculate roots: x1 = 1.0, x2 = -6.0.
Final Answer: x = 1.0, -6.0 |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 20^2 = c^2
2. Calculate the squares: 324 + 400 = c^2
3. Sum the squares: 724 = c^2
4. Take the square root of both sides: c = √724
Final A... |
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 +... |
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 ... |
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 =
[-5, 4]
[5, 4]
B =
[5, 0]
[-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) = (-5*5) + (4*-4) = -41
C12 = (A11 * B12) + (A12 * B22) = (-5*0) + (4*3) = 12
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (4*-4) = 9
C22 = (A21 * B12) + (A22 * B22) = (5*0) + (4*3) = 12
Result Ma... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 13, 8, 14, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 13 + 8 + 14 + 5) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, 2.8, -2.2, 3.8, -5.2]
Squared Differences: [0.64, 7.84, 4.84, 14.44, 27.04]
... |
Add the fractions 11/10 and 8/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 9, which is 90.
2. Convert 11/10 by multiplying top and bottom by 9: 99/90.
3. Convert 8/9 by multiplying top and bottom by 10: 80/90.
4. Add the numerators: 99 + 80 = 179.
5. The resulting fraction is 179/90.
Final Answer... |
Add the fractions 2/12 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 13, which is 156.
2. Convert 2/12 by multiplying top and bottom by 13: 26/156.
3. Convert 5/13 by multiplying top and bottom by 12: 60/156.
4. Add the numerators: 26 + 60 = 86.
5. The resulting fraction is 86/156.
6. Simpl... |
Compute the indefinite integral of f(x) = 8x^5. | To find the indefinite integral ∫ 8x^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: 8 / 6 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^5 dx = 4/3x^6 + C |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
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: 1x^2 + -4x + -7 = 0 | 1. Identify coefficients: a=1, b=-4, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -28 = 44.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √44) / 2.
5. Calculate roots: x1 = 5.3166247903554, x2 = -1.3166247903553998.
Final Answer... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
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 =
[0, -2]
[-3, 0]
B =
[4, 3]
[5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*4) + (-2*5) = -10
C12 = (A11 * B12) + (A12 * B22) = (0*3) + (-2*-2) = 4
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (0*5) = -12
C22 = (A21 * B12) + (A22 * B22) = (-3*3) + (0*-2) = -9
Result... |
Compute the quotient and remainder of 47464 divided by 51. Show the long division steps. | To divide 47464 by 51:
1. Determine how many times 51 goes into the leading digits of 47464.
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: 47464 / 51 = 930 with a remainder of 34.
Proof: (930 * 51) ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 4^2 = c^2
2. Calculate the squares: 196 + 16 = c^2
3. Sum the squares: 212 = c^2
4. Take the square root of both sides: c = √212
Final Ans... |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
Find the roots of the quadratic equation: 3x^2 + -7x + -10 = 0 | 1. Identify coefficients: a=3, b=-7, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -120 = 169.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √169) / 6.
5. Calculate roots: x1 = 3.3333333333333335, x2 = -1.0.
Final Answer: x = 3.... |
Find the roots of the quadratic equation: 3x^2 + -10x + -7 = 0 | 1. Identify coefficients: a=3, b=-10, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -84 = 184.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √184) / 6.
5. Calculate roots: x1 = 3.927443327708423, x2 = -0.5941099943750894.
Final... |
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 |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[2, -4]
B =
[1, 4]
[-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*1) + (-5*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (-1*4) + (-5*5) = -29
C21 = (A21 * B11) + (A22 * B21) = (2*1) + (-4*-1) = 6
C22 = (A21 * B12) + (A22 * B22) = (2*4) + (-4*5) = -12
Resul... |
Multiply the following 2x2 matrices:
A =
[1, -1]
[4, 0]
B =
[2, 3]
[-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*2) + (-1*-1) = 3
C12 = (A11 * B12) + (A12 * B22) = (1*3) + (-1*3) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (0*-1) = 8
C22 = (A21 * B12) + (A22 * B22) = (4*3) + (0*3) = 12
Result Matri... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 9^2 = c^2
2. Calculate the squares: 49 + 81 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answe... |
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 |
Add the fractions 8/6 and 1/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 2, which is 6.
2. Convert 8/6 by multiplying top and bottom by 1: 8/6.
3. Convert 1/2 by multiplying top and bottom by 3: 3/6.
4. Add the numerators: 8 + 3 = 11.
5. The resulting fraction is 11/6.
Final Answer: 11/6 |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 14^2 = c^2
2. Calculate the squares: 256 + 196 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[3, 3]
B =
[-1, -4]
[3, -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*-1) + (4*3) = 14
C12 = (A11 * B12) + (A12 * B22) = (-2*-4) + (4*-3) = -4
C21 = (A21 * B11) + (A22 * B21) = (3*-1) + (3*3) = 6
C22 = (A21 * B12) + (A22 * B22) = (3*-4) + (3*-3) = -21
Resul... |
Find the roots of the quadratic equation: 2x^2 + 7x + 10 = 0 | 1. Identify coefficients: a=2, b=7, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(2)(10) = 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) / 4.
Final Answer: x = -1.75 ± 1.3919410907075054i |
Multiply the following 2x2 matrices:
A =
[0, -4]
[4, -1]
B =
[0, -2]
[3, -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*0) + (-4*3) = -12
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (-4*-3) = 12
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (-1*3) = -3
C22 = (A21 * B12) + (A22 * B22) = (4*-2) + (-1*-3) = -5
Resu... |
Compute the quotient and remainder of 14612 divided by 97. Show the long division steps. | To divide 14612 by 97:
1. Determine how many times 97 goes into the leading digits of 14612.
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: 14612 / 97 = 150 with a remainder of 62.
Proof: (150 * 97) ... |
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 +... |
Add the fractions 13/10 and 9/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 4, which is 20.
2. Convert 13/10 by multiplying top and bottom by 2: 26/20.
3. Convert 9/4 by multiplying top and bottom by 5: 45/20.
4. Add the numerators: 26 + 45 = 71.
5. The resulting fraction is 71/20.
Final Answer: 7... |
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 +... |
Multiply the following 2x2 matrices:
A =
[3, 2]
[0, -2]
B =
[-2, -3]
[1, -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*-2) + (2*1) = -4
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (2*-4) = -17
C21 = (A21 * B11) + (A22 * B21) = (0*-2) + (-2*1) = -2
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (-2*-4) = 8
Resul... |
Find the roots of the quadratic equation: 4x^2 + -5x + 5 = 0 | 1. Identify coefficients: a=4, b=-5, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(4)(5) = 25 - 80 = -55.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √55i) / 8.
Final Answer: x = 0.625 ± 0.9270248108869579i |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
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 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 |
Multiply the following 2x2 matrices:
A =
[5, 4]
[4, -1]
B =
[0, 4]
[-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) = (5*0) + (4*-4) = -16
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (4*-3) = 8
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (-1*-4) = 4
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (-1*-3) = 19
Result M... |
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 |
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 |
Find the roots of the quadratic equation: 5x^2 + 6x + 1 = 0 | 1. Identify coefficients: a=5, b=6, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 20 = 16.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √16) / 10.
5. Calculate roots: x1 = -0.2, x2 = -1.0.
Final Answer: x = -0.2, -1.0 |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 3^2 = c^2
2. Calculate the squares: 225 + 9 = c^2
3. Sum the squares: 234 = c^2
4. Take the square root of both sides: c = √234
Final Answ... |
Find the roots of the quadratic equation: 2x^2 + -1x + -4 = 0 | 1. Identify coefficients: a=2, b=-1, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -32 = 33.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √33) / 4.
5. Calculate roots: x1 = 1.6861406616345072, x2 = -1.1861406616345072.
Final Answ... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 20^2 = c^2
2. Calculate the squares: 144 + 400 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
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) |
Compute the quotient and remainder of 48677 divided by 4. Show the long division steps. | To divide 48677 by 4:
1. Determine how many times 4 goes into the leading digits of 48677.
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: 48677 / 4 = 12169 with a remainder of 1.
Proof: (12169 * 4) +... |
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 = 17 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: 17^2 + 16^2 = c^2
2. Calculate the squares: 289 + 256 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
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: 3x^2 + -1x + 8 = 0 | 1. Identify coefficients: a=3, b=-1, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(3)(8) = 1 - 96 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √95i) / 6.
Final Answer: x = 0.16666666666666666 ± 1.6244657241348273i |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 19, 13, 16, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 19 + 13 + 16 + 17) / 5 = 82 / 5 = 16.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, 2.6, -3.4, -0.4, 0.6]
Squared Differences: [0.36, 6.76, 11.56, 0.16, 0.36]
... |
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 ... |
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