prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-5, 1]
[-4, -5]
B =
[-1, 2]
[-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) = (-5*-1) + (1*-2) = 3
C12 = (A11 * B12) + (A12 * B22) = (-5*2) + (1*-4) = -14
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (-5*-2) = 14
C22 = (A21 * B12) + (A22 * B22) = (-4*2) + (-5*-4) = 12
R... |
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) |
Multiply the following 2x2 matrices:
A =
[3, 4]
[3, -2]
B =
[2, 4]
[4, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*2) + (4*4) = 22
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (4*1) = 16
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-2*4) = -2
C22 = (A21 * B12) + (A22 * B22) = (3*4) + (-2*1) = 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 +... |
Find the roots of the quadratic equation: 7x^2 + -4x + 7 = 0 | 1. Identify coefficients: a=7, b=-4, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(7)(7) = 16 - 196 = -180.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √180i) / 14.
Final Answer: x = 0.2857142857142857 ± 0.9583148474999099i |
Multiply the following 2x2 matrices:
A =
[-2, 5]
[1, -4]
B =
[5, 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) = (-2*5) + (5*5) = 15
C12 = (A11 * B12) + (A12 * B22) = (-2*1) + (5*1) = 3
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-4*5) = -15
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (-4*1) = -3
Result Ma... |
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 |
Find the roots of the quadratic equation: 9x^2 + -2x + -9 = 0 | 1. Identify coefficients: a=9, b=-2, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -324 = 328.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √328) / 18.
5. Calculate roots: x1 = 1.117265015348602, x2 = -0.8950427931263797.
Final A... |
Compute the quotient and remainder of 5105 divided by 73. Show the long division steps. | To divide 5105 by 73:
1. Determine how many times 73 goes into the leading digits of 5105.
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: 5105 / 73 = 69 with a remainder of 68.
Proof: (69 * 73) + 68 ... |
Find the roots of the quadratic equation: 8x^2 + -6x + -8 = 0 | 1. Identify coefficients: a=8, b=-6, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -256 = 292.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √292) / 16.
5. Calculate roots: x1 = 1.4430004681646913, x2 = -0.6930004681646913.
Final... |
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) |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^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: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
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 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 tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 6x^2 + -1x + -3 = 0 | 1. Identify coefficients: a=6, b=-1, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -72 = 73.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √73) / 12.
5. Calculate roots: x1 = 0.7953336454431276, x2 = -0.6286669787764608.
Final Ans... |
Find the roots of the quadratic equation: 9x^2 + 8x + -2 = 0 | 1. Identify coefficients: a=9, b=8, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -72 = 136.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √136) / 18.
5. Calculate roots: x1 = 0.20343909942725563, x2 = -1.0923279883161445.
Final... |
Find the roots of the quadratic equation: 3x^2 + 0x + 9 = 0 | 1. Identify coefficients: a=3, b=0, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(3)(9) = 0 - 108 = -108.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √108i) / 6.
Final Answer: x = 0.0 ± 1.7320508075688774i |
Add the fractions 5/15 and 4/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 15, which is 15.
2. Convert 5/15 by multiplying top and bottom by 1: 5/15.
3. Convert 4/15 by multiplying top and bottom by 1: 4/15.
4. Add the numerators: 5 + 4 = 9.
5. The resulting fraction is 9/15.
6. Simplify by divid... |
Compute the quotient and remainder of 98851 divided by 9. Show the long division steps. | To divide 98851 by 9:
1. Determine how many times 9 goes into the leading digits of 98851.
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: 98851 / 9 = 10983 with a remainder of 4.
Proof: (10983 * 9) +... |
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 +... |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[1, -3]
B =
[5, 3]
[1, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*5) + (4*1) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (4*0) = -6
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-3*1) = 2
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-3*0) = 3
Result Matr... |
Multiply the following 2x2 matrices:
A =
[-5, 2]
[0, 1]
B =
[5, 3]
[3, 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) = (-5*5) + (2*3) = -19
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (2*1) = -13
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (1*3) = 3
C22 = (A21 * B12) + (A22 * B22) = (0*3) + (1*1) = 1
Result Matr... |
Compute the indefinite integral of f(x) = 1x^5. | To find the indefinite integral ∫ 1x^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: 1 / 6 = 1/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^5 dx = 1/6x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 3, 2, 18, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 3 + 2 + 18 + 9) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, -3.8, -4.8, 11.2, 2.2]
Squared Differences: [23.04, 14.44, 23.04, 125.44, 4.84]
... |
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(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 |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 1, 4, 9, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 1 + 4 + 9 + 6) / 5 = 26 / 5 = 5.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, -4.2, -1.2, 3.8, 0.8]
Squared Differences: [0.64, 17.64, 1.44, 14.44, 0.64]
Sum... |
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 |
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 +... |
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 ... |
Find the roots of the quadratic equation: 9x^2 + -7x + 7 = 0 | 1. Identify coefficients: a=9, b=-7, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(9)(7) = 49 - 252 = -203.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √203i) / 18.
Final Answer: x = 0.3888888888888889 ± 0.7915448249319448i |
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 |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 4, 1, 3, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 4 + 1 + 3 + 6) / 5 = 21 / 5 = 4.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -0.2, -3.2, -1.2, 1.8]
Squared Differences: [7.84, 0.04, 10.24, 1.44, 3.24]
Sum... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 17, 2, 10, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 17 + 2 + 10 + 3) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.6, 6.6, -8.4, -0.4, -7.4]
Squared Differences: [92.16, 43.56, 70.56, 0.16, 54.76]... |
Multiply the following 2x2 matrices:
A =
[-2, 3]
[-2, -4]
B =
[5, 2]
[4, 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*5) + (3*4) = 2
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (3*0) = -4
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (-4*4) = -26
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (-4*0) = -4
Result ... |
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) |
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) |
Compute the quotient and remainder of 47184 divided by 13. Show the long division steps. | To divide 47184 by 13:
1. Determine how many times 13 goes into the leading digits of 47184.
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: 47184 / 13 = 3629 with a remainder of 7.
Proof: (3629 * 13)... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 18^2 = c^2
2. Calculate the squares: 25 + 324 = c^2
3. Sum the squares: 349 = c^2
4. Take the square root of both sides: c = √349
Final Ans... |
Compute the quotient and remainder of 3535 divided by 66. Show the long division steps. | To divide 3535 by 66:
1. Determine how many times 66 goes into the leading digits of 3535.
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: 3535 / 66 = 53 with a remainder of 37.
Proof: (53 * 66) + 37 ... |
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 |
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: [9, 12, 5, 19, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 12 + 5 + 19 + 14) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 0.2, -6.8, 7.2, 2.2]
Squared Differences: [7.84, 0.04, 46.24, 51.84, 4.84]
... |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 6^2 + 19^2 = c^2
2. Calculate the squares: 36 + 361 = c^2
3. Sum the squares: 397 = c^2
4. Take the square root of both sides: c = √397
Final Ans... |
Add the fractions 6/4 and 13/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 11, which is 44.
2. Convert 6/4 by multiplying top and bottom by 11: 66/44.
3. Convert 13/11 by multiplying top and bottom by 4: 52/44.
4. Add the numerators: 66 + 52 = 118.
5. The resulting fraction is 118/44.
6. Simplify ... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[2, -5]
B =
[5, -5]
[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) = (1*5) + (5*1) = 10
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (5*-4) = -25
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (-5*1) = 5
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-5*-4) = 10
Result ... |
Find the derivative of f(x) = (5x + 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 = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 23326 divided by 86. Show the long division steps. | To divide 23326 by 86:
1. Determine how many times 86 goes into the leading digits of 23326.
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: 23326 / 86 = 271 with a remainder of 20.
Proof: (271 * 86) ... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[4, 4]
B =
[-2, -1]
[-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*-2) + (3*-2) = -10
C12 = (A11 * B12) + (A12 * B22) = (2*-1) + (3*-5) = -17
C21 = (A21 * B11) + (A22 * B21) = (4*-2) + (4*-2) = -16
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (4*-5) = -24
R... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 17^2 = c^2
2. Calculate the squares: 16 + 289 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 20^2 + 19^2 = c^2
2. Calculate the squares: 400 + 361 = c^2
3. Sum the squares: 761 = c^2
4. Take the square root of both sides: c = √761
Final A... |
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) |
Multiply the following 2x2 matrices:
A =
[5, -2]
[4, -4]
B =
[-5, 2]
[0, 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) = (5*-5) + (-2*0) = -25
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (-2*5) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (-4*0) = -20
C22 = (A21 * B12) + (A22 * B22) = (4*2) + (-4*5) = -12
Resul... |
Compute the indefinite integral of f(x) = 4x^2. | To find the indefinite integral ∫ 4x^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: 4 / 3 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^2 dx = 4/3x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 6, 18, 4, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 6 + 18 + 4 + 14) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, -2.6, 9.4, -4.6, 5.4]
Squared Differences: [57.76, 6.76, 88.36, 21.16, 29.16]
... |
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) |
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... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 13^2 = c^2
2. Calculate the squares: 121 + 169 = c^2
3. Sum the squares: 290 = c^2
4. Take the square root of both sides: c = √290
Final A... |
Multiply the following 2x2 matrices:
A =
[2, 4]
[-2, -5]
B =
[4, 1]
[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) = (2*4) + (4*4) = 24
C12 = (A11 * B12) + (A12 * B22) = (2*1) + (4*-4) = -14
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (-5*4) = -28
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (-5*-4) = 18
Resul... |
Compute the quotient and remainder of 12594 divided by 67. Show the long division steps. | To divide 12594 by 67:
1. Determine how many times 67 goes into the leading digits of 12594.
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: 12594 / 67 = 187 with a remainder of 65.
Proof: (187 * 67) ... |
Find the roots of the quadratic equation: 2x^2 + -3x + -2 = 0 | 1. Identify coefficients: a=2, b=-3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -16 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √25) / 4.
5. Calculate roots: x1 = 2.0, x2 = -0.5.
Final Answer: x = 2.0, -0.5 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 11^2 = c^2
2. Calculate the squares: 400 + 121 = c^2
3. Sum the squares: 521 = c^2
4. Take the square root of both sides: c = √521
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 20^2 = c^2
2. Calculate the squares: 256 + 400 = c^2
3. Sum the squares: 656 = c^2
4. Take the square root of both sides: c = √656
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 sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(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) |
Find the roots of the quadratic equation: 5x^2 + 6x + -7 = 0 | 1. Identify coefficients: a=5, b=6, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -140 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √176) / 10.
5. Calculate roots: x1 = 0.7266499161421599, x2 = -1.92664991614216.
Final A... |
Compute the indefinite integral of f(x) = 2x^1. | To find the indefinite integral ∫ 2x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^2 + C |
Evaluate sin(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 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) |
Add the fractions 3/7 and 8/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 10, which is 70.
2. Convert 3/7 by multiplying top and bottom by 10: 30/70.
3. Convert 8/10 by multiplying top and bottom by 7: 56/70.
4. Add the numerators: 30 + 56 = 86.
5. The resulting fraction is 86/70.
6. Simplify by ... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 19, 19, 9, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 19 + 19 + 9 + 9) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, 5.6, 5.6, -4.4, -4.4]
Squared Differences: [5.76, 31.36, 31.36, 19.36, 19.36]... |
Find the roots of the quadratic equation: 8x^2 + -5x + -7 = 0 | 1. Identify coefficients: a=8, b=-5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -224 = 249.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √249) / 16.
5. Calculate roots: x1 = 1.2987333648787187, x2 = -0.6737333648787187.
Final... |
Add the fractions 13/10 and 13/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 13, which is 130.
2. Convert 13/10 by multiplying top and bottom by 13: 169/130.
3. Convert 13/13 by multiplying top and bottom by 10: 130/130.
4. Add the numerators: 169 + 130 = 299.
5. The resulting fraction is 299/130.
... |
Add the fractions 6/8 and 13/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 14, which is 56.
2. Convert 6/8 by multiplying top and bottom by 7: 42/56.
3. Convert 13/14 by multiplying top and bottom by 4: 52/56.
4. Add the numerators: 42 + 52 = 94.
5. The resulting fraction is 94/56.
6. Simplify by ... |
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 10/13 and 14/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 10/13 by multiplying top and bottom by 1: 10/13.
3. Convert 14/13 by multiplying top and bottom by 1: 14/13.
4. Add the numerators: 10 + 14 = 24.
5. The resulting fraction is 24/13.
Final Answer... |
Multiply the following 2x2 matrices:
A =
[-4, -4]
[-2, 4]
B =
[4, 0]
[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) = (-4*4) + (-4*1) = -20
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (-4*-5) = 20
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (4*1) = -4
C22 = (A21 * B12) + (A22 * B22) = (-2*0) + (4*-5) = -20
Res... |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[-3, 5]
B =
[4, 1]
[-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) = (-5*4) + (-2*-5) = -10
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (-2*-5) = 5
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (5*-5) = -37
C22 = (A21 * B12) + (A22 * B22) = (-3*1) + (5*-5) = -28
R... |
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 +... |
Add the fractions 14/6 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 4, which is 12.
2. Convert 14/6 by multiplying top and bottom by 2: 28/12.
3. Convert 6/4 by multiplying top and bottom by 3: 18/12.
4. Add the numerators: 28 + 18 = 46.
5. The resulting fraction is 46/12.
6. Simplify by di... |
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 |
Compute the quotient and remainder of 64867 divided by 13. Show the long division steps. | To divide 64867 by 13:
1. Determine how many times 13 goes into the leading digits of 64867.
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: 64867 / 13 = 4989 with a remainder of 10.
Proof: (4989 * 13... |
Find the derivative of f(x) = (3x + 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 = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Compute the quotient and remainder of 61779 divided by 94. Show the long division steps. | To divide 61779 by 94:
1. Determine how many times 94 goes into the leading digits of 61779.
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: 61779 / 94 = 657 with a remainder of 21.
Proof: (657 * 94) ... |
Find the roots of the quadratic equation: 5x^2 + 2x + -1 = 0 | 1. Identify coefficients: a=5, b=2, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -20 = 24.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √24) / 10.
5. Calculate roots: x1 = 0.2898979485566356, x2 = -0.6898979485566356.
Final Ans... |
Add the fractions 11/4 and 13/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 15, which is 60.
2. Convert 11/4 by multiplying top and bottom by 15: 165/60.
3. Convert 13/15 by multiplying top and bottom by 4: 52/60.
4. Add the numerators: 165 + 52 = 217.
5. The resulting fraction is 217/60.
Final Ans... |
Add the fractions 9/8 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 6, which is 24.
2. Convert 9/8 by multiplying top and bottom by 3: 27/24.
3. Convert 3/6 by multiplying top and bottom by 4: 12/24.
4. Add the numerators: 27 + 12 = 39.
5. The resulting fraction is 39/24.
6. Simplify by div... |
Add the fractions 12/15 and 15/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 10, which is 30.
2. Convert 12/15 by multiplying top and bottom by 2: 24/30.
3. Convert 15/10 by multiplying top and bottom by 3: 45/30.
4. Add the numerators: 24 + 45 = 69.
5. The resulting fraction is 69/30.
6. Simplify ... |
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: 9x^2 + 1x + -6 = 0 | 1. Identify coefficients: a=9, b=1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -216 = 217.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √217) / 18.
5. Calculate roots: x1 = 0.7628288812586796, x2 = -0.8739399923697908.
Final ... |
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 indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^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: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Compute the quotient and remainder of 17017 divided by 99. Show the long division steps. | To divide 17017 by 99:
1. Determine how many times 99 goes into the leading digits of 17017.
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: 17017 / 99 = 171 with a remainder of 88.
Proof: (171 * 99) ... |
Compute the quotient and remainder of 7219 divided by 22. Show the long division steps. | To divide 7219 by 22:
1. Determine how many times 22 goes into the leading digits of 7219.
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: 7219 / 22 = 328 with a remainder of 3.
Proof: (328 * 22) + 3 ... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 18^2 = c^2
2. Calculate the squares: 196 + 324 = c^2
3. Sum the squares: 520 = c^2
4. Take the square root of both sides: c = √520
Final A... |
Add the fractions 3/10 and 12/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 14, which is 70.
2. Convert 3/10 by multiplying top and bottom by 7: 21/70.
3. Convert 12/14 by multiplying top and bottom by 5: 60/70.
4. Add the numerators: 21 + 60 = 81.
5. The resulting fraction is 81/70.
Final Answer:... |
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