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In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 13^2 = c^2
2. Calculate the squares: 9 + 169 = c^2
3. Sum the squares: 178 = c^2
4. Take the square root of both sides: c = √178
Final Answ... |
Find the roots of the quadratic equation: 2x^2 + 7x + 0 = 0 | 1. Identify coefficients: a=2, b=7, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 0 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √49) / 4.
5. Calculate roots: x1 = 0.0, x2 = -3.5.
Final Answer: x = 0.0, -3.5 |
Find the roots of the quadratic equation: 6x^2 + 6x + 10 = 0 | 1. Identify coefficients: a=6, b=6, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(6)(10) = 36 - 240 = -204.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √204i) / 12.
Final Answer: x = -0.5 ± 1.1902380714238083i |
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 +... |
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) |
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 |
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 ... |
Compute the quotient and remainder of 49274 divided by 44. Show the long division steps. | To divide 49274 by 44:
1. Determine how many times 44 goes into the leading digits of 49274.
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: 49274 / 44 = 1119 with a remainder of 38.
Proof: (1119 * 44... |
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 |
Find the roots of the quadratic equation: 9x^2 + -4x + 9 = 0 | 1. Identify coefficients: a=9, b=-4, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(9)(9) = 16 - 324 = -308.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √308i) / 18.
Final Answer: x = 0.2222222222222222 ± 0.9749960430435691i |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 15, 20, 17, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 15 + 20 + 17 + 16) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-11.2, 0.8, 5.8, 2.8, 1.8]
Squared Differences: [125.44, 0.64, 33.64, 7.84, 3.24]
... |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Add the fractions 11/4 and 8/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 10, which is 20.
2. Convert 11/4 by multiplying top and bottom by 5: 55/20.
3. Convert 8/10 by multiplying top and bottom by 2: 16/20.
4. Add the numerators: 55 + 16 = 71.
5. The resulting fraction is 71/20.
Final Answer: 7... |
Add the fractions 5/11 and 12/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 3, which is 33.
2. Convert 5/11 by multiplying top and bottom by 3: 15/33.
3. Convert 12/3 by multiplying top and bottom by 11: 132/33.
4. Add the numerators: 15 + 132 = 147.
5. The resulting fraction is 147/33.
6. Simplif... |
Find the roots of the quadratic equation: 4x^2 + 1x + -2 = 0 | 1. Identify coefficients: a=4, b=1, c=-2.
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) / 8.
5. Calculate roots: x1 = 0.5930703308172536, x2 = -0.8430703308172536.
Final Answ... |
Find the roots of the quadratic equation: 4x^2 + 10x + -2 = 0 | 1. Identify coefficients: a=4, b=10, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -32 = 132.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √132) / 8.
5. Calculate roots: x1 = 0.18614066163450715, x2 = -2.686140661634507.
Fina... |
Add the fractions 9/2 and 11/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 8, which is 8.
2. Convert 9/2 by multiplying top and bottom by 4: 36/8.
3. Convert 11/8 by multiplying top and bottom by 1: 11/8.
4. Add the numerators: 36 + 11 = 47.
5. The resulting fraction is 47/8.
Final Answer: 47/8 |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 8^2 = c^2
2. Calculate the squares: 16 + 64 = c^2
3. Sum the squares: 80 = c^2
4. Take the square root of both sides: c = √80
Final Answer:... |
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 |
Multiply the following 2x2 matrices:
A =
[1, 5]
[-4, 0]
B =
[1, 1]
[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) = (1*1) + (5*2) = 11
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (5*-4) = -19
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (0*2) = -4
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (0*-4) = -4
Result M... |
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 ... |
Add the fractions 6/7 and 8/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 8, which is 56.
2. Convert 6/7 by multiplying top and bottom by 8: 48/56.
3. Convert 8/8 by multiplying top and bottom by 7: 56/56.
4. Add the numerators: 48 + 56 = 104.
5. The resulting fraction is 104/56.
6. Simplify by d... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 9, 18, 17, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 9 + 18 + 17 + 6) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, -2.2, 6.8, 5.8, -5.2]
Squared Differences: [27.04, 4.84, 46.24, 33.64, 27.04]
... |
Compute the quotient and remainder of 30924 divided by 42. Show the long division steps. | To divide 30924 by 42:
1. Determine how many times 42 goes into the leading digits of 30924.
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: 30924 / 42 = 736 with a remainder of 12.
Proof: (736 * 42) ... |
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 +... |
Find the roots of the quadratic equation: 1x^2 + 2x + 3 = 0 | 1. Identify coefficients: a=1, b=2, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(1)(3) = 4 - 12 = -8.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √8i) / 2.
Final Answer: x = -1.0 ± 1.4142135623730951i |
Find the roots of the quadratic equation: 10x^2 + -7x + -8 = 0 | 1. Identify coefficients: a=10, b=-7, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -320 = 369.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √369) / 20.
5. Calculate roots: x1 = 1.3104686356149273, x2 = -0.6104686356149273.
Fina... |
Compute the quotient and remainder of 29993 divided by 79. Show the long division steps. | To divide 29993 by 79:
1. Determine how many times 79 goes into the leading digits of 29993.
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: 29993 / 79 = 379 with a remainder of 52.
Proof: (379 * 79) ... |
Compute the quotient and remainder of 56213 divided by 12. Show the long division steps. | To divide 56213 by 12:
1. Determine how many times 12 goes into the leading digits of 56213.
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: 56213 / 12 = 4684 with a remainder of 5.
Proof: (4684 * 12)... |
Add the fractions 3/13 and 7/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 12, which is 156.
2. Convert 3/13 by multiplying top and bottom by 12: 36/156.
3. Convert 7/12 by multiplying top and bottom by 13: 91/156.
4. Add the numerators: 36 + 91 = 127.
5. The resulting fraction is 127/156.
Final ... |
Find the roots of the quadratic equation: 8x^2 + 1x + 9 = 0 | 1. Identify coefficients: a=8, b=1, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(8)(9) = 1 - 288 = -287.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √287i) / 16.
Final Answer: x = -0.0625 ± 1.0588171466310885i |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 51115 divided by 58. Show the long division steps. | To divide 51115 by 58:
1. Determine how many times 58 goes into the leading digits of 51115.
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: 51115 / 58 = 881 with a remainder of 17.
Proof: (881 * 58) ... |
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 |
Compute the quotient and remainder of 38664 divided by 4. Show the long division steps. | To divide 38664 by 4:
1. Determine how many times 4 goes into the leading digits of 38664.
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: 38664 / 4 = 9666 with a remainder of 0.
Proof: (9666 * 4) + 0... |
Compute the quotient and remainder of 93237 divided by 3. Show the long division steps. | To divide 93237 by 3:
1. Determine how many times 3 goes into the leading digits of 93237.
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: 93237 / 3 = 31079 with a remainder of 0.
Proof: (31079 * 3) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 9, 1, 18, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 9 + 1 + 18 + 7) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -0.6, -8.6, 8.4, -2.6]
Squared Differences: [11.56, 0.36, 73.96, 70.56, 6.76]
... |
Compute the quotient and remainder of 72205 divided by 16. Show the long division steps. | To divide 72205 by 16:
1. Determine how many times 16 goes into the leading digits of 72205.
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: 72205 / 16 = 4512 with a remainder of 13.
Proof: (4512 * 16... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 6, 7, 10, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 6 + 7 + 10 + 15) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, -2.0, -1.0, 2.0, 7.0]
Squared Differences: [36.0, 4.0, 1.0, 4.0, 49.0]
Sum o... |
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 |
Compute the quotient and remainder of 7347 divided by 17. Show the long division steps. | To divide 7347 by 17:
1. Determine how many times 17 goes into the leading digits of 7347.
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: 7347 / 17 = 432 with a remainder of 3.
Proof: (432 * 17) + 3 ... |
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 ... |
Compute the indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^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: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + C |
Compute the quotient and remainder of 22570 divided by 38. Show the long division steps. | To divide 22570 by 38:
1. Determine how many times 38 goes into the leading digits of 22570.
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: 22570 / 38 = 593 with a remainder of 36.
Proof: (593 * 38) ... |
Add the fractions 6/14 and 2/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 6/14 by multiplying top and bottom by 3: 18/42.
3. Convert 2/3 by multiplying top and bottom by 14: 28/42.
4. Add the numerators: 18 + 28 = 46.
5. The resulting fraction is 46/42.
6. Simplify by ... |
Compute the quotient and remainder of 93141 divided by 59. Show the long division steps. | To divide 93141 by 59:
1. Determine how many times 59 goes into the leading digits of 93141.
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: 93141 / 59 = 1578 with a remainder of 39.
Proof: (1578 * 59... |
Compute the quotient and remainder of 52785 divided by 73. Show the long division steps. | To divide 52785 by 73:
1. Determine how many times 73 goes into the leading digits of 52785.
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: 52785 / 73 = 723 with a remainder of 6.
Proof: (723 * 73) +... |
Compute the quotient and remainder of 44723 divided by 63. Show the long division steps. | To divide 44723 by 63:
1. Determine how many times 63 goes into the leading digits of 44723.
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: 44723 / 63 = 709 with a remainder of 56.
Proof: (709 * 63) ... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Compute the quotient and remainder of 45801 divided by 71. Show the long division steps. | To divide 45801 by 71:
1. Determine how many times 71 goes into the leading digits of 45801.
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: 45801 / 71 = 645 with a remainder of 6.
Proof: (645 * 71) +... |
Add the fractions 11/3 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 3, which is 3.
2. Convert 11/3 by multiplying top and bottom by 1: 11/3.
3. Convert 14/3 by multiplying top and bottom by 1: 14/3.
4. Add the numerators: 11 + 14 = 25.
5. The resulting fraction is 25/3.
Final Answer: 25/3 |
Multiply the following 2x2 matrices:
A =
[5, 4]
[-2, -4]
B =
[-4, 4]
[-5, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-4) + (4*-5) = -40
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (4*-3) = 8
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (-4*-5) = 28
C22 = (A21 * B12) + (A22 * B22) = (-2*4) + (-4*-3) = 4
Resu... |
Find the roots of the quadratic equation: 5x^2 + 0x + 4 = 0 | 1. Identify coefficients: a=5, b=0, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(5)(4) = 0 - 80 = -80.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √80i) / 10.
Final Answer: x = 0.0 ± 0.894427190999916i |
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 |
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) |
Find the roots of the quadratic equation: 8x^2 + -1x + -9 = 0 | 1. Identify coefficients: a=8, b=-1, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -288 = 289.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √289) / 16.
5. Calculate roots: x1 = 1.125, x2 = -1.0.
Final Answer: x = 1.125, -1.0 |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^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: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
Add the fractions 14/5 and 1/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 7, which is 35.
2. Convert 14/5 by multiplying top and bottom by 7: 98/35.
3. Convert 1/7 by multiplying top and bottom by 5: 5/35.
4. Add the numerators: 98 + 5 = 103.
5. The resulting fraction is 103/35.
Final Answer: 103... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 12, 20, 5, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 12 + 20 + 5 + 15) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, 0.2, 8.2, -6.8, 3.2]
Squared Differences: [23.04, 0.04, 67.24, 46.24, 10.24]
... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Evaluate 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 roots of the quadratic equation: 7x^2 + 3x + -10 = 0 | 1. Identify coefficients: a=7, b=3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -280 = 289.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √289) / 14.
5. Calculate roots: x1 = 1.0, x2 = -1.4285714285714286.
Final Answer: x = 1.... |
Compute the indefinite integral of f(x) = 3x^4. | To find the indefinite integral ∫ 3x^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: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
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 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 ... |
Compute the quotient and remainder of 7210 divided by 64. Show the long division steps. | To divide 7210 by 64:
1. Determine how many times 64 goes into the leading digits of 7210.
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: 7210 / 64 = 112 with a remainder of 42.
Proof: (112 * 64) + 4... |
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 ... |
Multiply the following 2x2 matrices:
A =
[0, 1]
[-4, -3]
B =
[-5, 5]
[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) = (0*-5) + (1*4) = 4
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (1*-4) = -4
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (-3*4) = 8
C22 = (A21 * B12) + (A22 * B22) = (-4*5) + (-3*-4) = -8
Result ... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 2, 12, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 2 + 12 + 9 + 18) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.0, -7.0, 3.0, 0.0, 9.0]
Squared Differences: [25.0, 49.0, 9.0, 0.0, 81.0]
Sum o... |
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) |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 42273 divided by 45. Show the long division steps. | To divide 42273 by 45:
1. Determine how many times 45 goes into the leading digits of 42273.
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: 42273 / 45 = 939 with a remainder of 18.
Proof: (939 * 45) ... |
Multiply the following 2x2 matrices:
A =
[2, 4]
[-3, -3]
B =
[1, 5]
[3, 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) + (4*3) = 14
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (4*5) = 30
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (-3*3) = -12
C22 = (A21 * B12) + (A22 * B22) = (-3*5) + (-3*5) = -30
Result ... |
Find the roots of the quadratic equation: 1x^2 + -8x + 9 = 0 | 1. Identify coefficients: a=1, b=-8, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - 36 = 28.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √28) / 2.
5. Calculate roots: x1 = 6.645751311064591, x2 = 1.3542486889354093.
Final Answer:... |
Compute the quotient and remainder of 26695 divided by 5. Show the long division steps. | To divide 26695 by 5:
1. Determine how many times 5 goes into the leading digits of 26695.
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: 26695 / 5 = 5339 with a remainder of 0.
Proof: (5339 * 5) + 0... |
Compute the quotient and remainder of 24913 divided by 94. Show the long division steps. | To divide 24913 by 94:
1. Determine how many times 94 goes into the leading digits of 24913.
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: 24913 / 94 = 265 with a remainder of 3.
Proof: (265 * 94) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 17, 6, 3, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 17 + 6 + 3 + 14) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 8.8, -2.2, -5.2, 5.8]
Squared Differences: [51.84, 77.44, 4.84, 27.04, 33.64]
... |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[-2, -3]
B =
[-4, -1]
[5, 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) = (-4*-4) + (-2*5) = 6
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (-2*1) = 2
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (-3*5) = -7
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (-3*1) = -1
Res... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 7^2 = c^2
2. Calculate the squares: 289 + 49 = c^2
3. Sum the squares: 338 = c^2
4. Take the square root of both sides: c = √338
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-3, -2]
[2, -4]
B =
[3, -3]
[-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) + (-2*-4) = -1
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (-2*4) = 1
C21 = (A21 * B11) + (A22 * B21) = (2*3) + (-4*-4) = 22
C22 = (A21 * B12) + (A22 * B22) = (2*-3) + (-4*4) = -22
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 6, 2, 1, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 6 + 2 + 1 + 16) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.4, -1.6, -5.6, -6.6, 8.4]
Squared Differences: [29.16, 2.56, 31.36, 43.56, 70.56]
... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 17^2 = c^2
2. Calculate the squares: 289 + 289 = c^2
3. Sum the squares: 578 = c^2
4. Take the square root of both sides: c = √578
Final A... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 4, 1, 3, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 4 + 1 + 3 + 12) / 5 = 29 / 5 = 5.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, -1.8, -4.8, -2.8, 6.2]
Squared Differences: [10.24, 3.24, 23.04, 7.84, 38.44]
... |
Multiply the following 2x2 matrices:
A =
[-5, -3]
[-5, 5]
B =
[0, -1]
[2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*0) + (-3*2) = -6
C12 = (A11 * B12) + (A12 * B22) = (-5*-1) + (-3*3) = -4
C21 = (A21 * B11) + (A22 * B21) = (-5*0) + (5*2) = 10
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (5*3) = 20
Resul... |
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... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 19, 5, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 19 + 5 + 9 + 18) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, 5.4, -8.6, -4.6, 4.4]
Squared Differences: [11.56, 29.16, 73.96, 21.16, 19.36]... |
Compute the quotient and remainder of 1535 divided by 99. Show the long division steps. | To divide 1535 by 99:
1. Determine how many times 99 goes into the leading digits of 1535.
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: 1535 / 99 = 15 with a remainder of 50.
Proof: (15 * 99) + 50 ... |
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 9/3 and 7/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 11, which is 33.
2. Convert 9/3 by multiplying top and bottom by 11: 99/33.
3. Convert 7/11 by multiplying top and bottom by 3: 21/33.
4. Add the numerators: 99 + 21 = 120.
5. The resulting fraction is 120/33.
6. Simplify b... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 20, 7, 10, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 20 + 7 + 10 + 5) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, 8.8, -4.2, -1.2, -6.2]
Squared Differences: [7.84, 77.44, 17.64, 1.44, 38.44]
... |
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 = 5 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: 5^2 + 11^2 = c^2
2. Calculate the squares: 25 + 121 = c^2
3. Sum the squares: 146 = c^2
4. Take the square root of both sides: c = √146
Final Ans... |
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) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
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: [17, 3, 13, 15, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 3 + 13 + 15 + 6) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.2, -7.8, 2.2, 4.2, -4.8]
Squared Differences: [38.44, 60.84, 4.84, 17.64, 23.04]
... |
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