prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[1, 2]
[-4, 5]
B =
[-5, -1]
[-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) = (1*-5) + (2*-3) = -11
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (2*-5) = -11
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (5*-3) = 5
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (5*-5) = -21
R... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Add the fractions 10/11 and 15/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 4, which is 44.
2. Convert 10/11 by multiplying top and bottom by 4: 40/44.
3. Convert 15/4 by multiplying top and bottom by 11: 165/44.
4. Add the numerators: 40 + 165 = 205.
5. The resulting fraction is 205/44.
Final Ans... |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 6x^2 + 2x + 6 = 0 | 1. Identify coefficients: a=6, b=2, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(6)(6) = 4 - 144 = -140.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √140i) / 12.
Final Answer: x = -0.16666666666666666 ± 0.9860132971832694i |
Find the roots of the quadratic equation: 6x^2 + -8x + -4 = 0 | 1. Identify coefficients: a=6, b=-8, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -96 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √160) / 12.
5. Calculate roots: x1 = 1.7207592200561266, x2 = -0.38742588672279316.
Final... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (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(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 12894 divided by 91. Show the long division steps. | To divide 12894 by 91:
1. Determine how many times 91 goes into the leading digits of 12894.
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: 12894 / 91 = 141 with a remainder of 63.
Proof: (141 * 91) ... |
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 |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Multiply the following 2x2 matrices:
A =
[1, 4]
[-2, 2]
B =
[0, -3]
[4, 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) = (1*0) + (4*4) = 16
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (4*2) = 5
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (2*4) = 8
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (2*2) = 10
Result Matr... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 19, 13, 14, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 19 + 13 + 14 + 15) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.0, 6.0, 0.0, 1.0, 2.0]
Squared Differences: [81.0, 36.0, 0.0, 1.0, 4.0]
Sum ... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 3x^2 + -8x + -5 = 0 | 1. Identify coefficients: a=3, b=-8, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -60 = 124.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √124) / 6.
5. Calculate roots: x1 = 3.1892547876100075, x2 = -0.5225881209433405.
Final A... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 3x^2 + -4x + 1 = 0 | 1. Identify coefficients: a=3, b=-4, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - 12 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √4) / 6.
5. Calculate roots: x1 = 1.0, x2 = 0.3333333333333333.
Final Answer: x = 1.0, 0.3333... |
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) |
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) |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 13^2 = c^2
2. Calculate the squares: 196 + 169 = c^2
3. Sum the squares: 365 = c^2
4. Take the square root of both sides: c = √365
Final A... |
Compute the quotient and remainder of 26226 divided by 30. Show the long division steps. | To divide 26226 by 30:
1. Determine how many times 30 goes into the leading digits of 26226.
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: 26226 / 30 = 874 with a remainder of 6.
Proof: (874 * 30) +... |
Find the roots of the quadratic equation: 7x^2 + 7x + -7 = 0 | 1. Identify coefficients: a=7, b=7, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -196 = 245.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √245) / 14.
5. Calculate roots: x1 = 0.6180339887498949, x2 = -1.618033988749895.
Final ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 13, 13, 3, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 13 + 13 + 3 + 8) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, 2.6, 2.6, -7.4, -2.4]
Squared Differences: [21.16, 6.76, 6.76, 54.76, 5.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 3, 14, 13, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 3 + 14 + 13 + 9) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, -7.4, 3.6, 2.6, -1.4]
Squared Differences: [6.76, 54.76, 12.96, 6.76, 1.96]
... |
Add the fractions 2/3 and 14/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 9, which is 9.
2. Convert 2/3 by multiplying top and bottom by 3: 6/9.
3. Convert 14/9 by multiplying top and bottom by 1: 14/9.
4. Add the numerators: 6 + 14 = 20.
5. The resulting fraction is 20/9.
Final Answer: 20/9 |
Compute the quotient and remainder of 99172 divided by 72. Show the long division steps. | To divide 99172 by 72:
1. Determine how many times 72 goes into the leading digits of 99172.
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: 99172 / 72 = 1377 with a remainder of 28.
Proof: (1377 * 72... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 12^2 = c^2
2. Calculate the squares: 289 + 144 = c^2
3. Sum the squares: 433 = c^2
4. Take the square root of both sides: c = √433
Final A... |
Add the fractions 3/14 and 1/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 3/14 by multiplying top and bottom by 1: 3/14.
3. Convert 1/14 by multiplying top and bottom by 1: 1/14.
4. Add the numerators: 3 + 1 = 4.
5. The resulting fraction is 4/14.
6. Simplify by divid... |
Add the fractions 8/5 and 2/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 6, which is 30.
2. Convert 8/5 by multiplying top and bottom by 6: 48/30.
3. Convert 2/6 by multiplying top and bottom by 5: 10/30.
4. Add the numerators: 48 + 10 = 58.
5. The resulting fraction is 58/30.
6. Simplify by div... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 7x^2 + -1x + 7 = 0 | 1. Identify coefficients: a=7, b=-1, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(7)(7) = 1 - 196 = -195.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √195i) / 14.
Final Answer: x = 0.07142857142857142 ± 0.9974457174120672i |
Add the fractions 9/13 and 3/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 7, which is 91.
2. Convert 9/13 by multiplying top and bottom by 7: 63/91.
3. Convert 3/7 by multiplying top and bottom by 13: 39/91.
4. Add the numerators: 63 + 39 = 102.
5. The resulting fraction is 102/91.
Final Answer:... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 10^2 = c^2
2. Calculate the squares: 289 + 100 = c^2
3. Sum the squares: 389 = c^2
4. Take the square root of both sides: c = √389
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) |
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 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 +... |
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) |
Multiply the following 2x2 matrices:
A =
[-3, 3]
[-4, -2]
B =
[-3, 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) = (-3*-3) + (3*-2) = 3
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (3*5) = 12
C21 = (A21 * B11) + (A22 * B21) = (-4*-3) + (-2*-2) = 16
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (-2*5) = -14
Res... |
Add the fractions 7/8 and 5/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 7/8 by multiplying top and bottom by 1: 7/8.
3. Convert 5/8 by multiplying top and bottom by 1: 5/8.
4. Add the numerators: 7 + 5 = 12.
5. The resulting fraction is 12/8.
6. Simplify by dividing nu... |
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 |
Compute the quotient and remainder of 42147 divided by 54. Show the long division steps. | To divide 42147 by 54:
1. Determine how many times 54 goes into the leading digits of 42147.
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: 42147 / 54 = 780 with a remainder of 27.
Proof: (780 * 54) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 13, 7, 17, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 13 + 7 + 17 + 2) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.4, 1.4, -4.6, 5.4, -9.6]
Squared Differences: [54.76, 1.96, 21.16, 29.16, 92.16]
... |
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 +... |
Find the roots of the quadratic equation: 1x^2 + 5x + -10 = 0 | 1. Identify coefficients: a=1, b=5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -40 = 65.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √65) / 2.
5. Calculate roots: x1 = 1.5311288741492746, x2 = -6.531128874149275.
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 8^2 = c^2
2. Calculate the squares: 361 + 64 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
Compute the quotient and remainder of 35380 divided by 68. Show the long division steps. | To divide 35380 by 68:
1. Determine how many times 68 goes into the leading digits of 35380.
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: 35380 / 68 = 520 with a remainder of 20.
Proof: (520 * 68) ... |
Add the fractions 1/3 and 11/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 2, which is 6.
2. Convert 1/3 by multiplying top and bottom by 2: 2/6.
3. Convert 11/2 by multiplying top and bottom by 3: 33/6.
4. Add the numerators: 2 + 33 = 35.
5. The resulting fraction is 35/6.
Final Answer: 35/6 |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 6^2 = c^2
2. Calculate the squares: 100 + 36 = c^2
3. Sum the squares: 136 = c^2
4. Take the square root of both sides: c = √136
Final Ans... |
Compute the quotient and remainder of 89826 divided by 45. Show the long division steps. | To divide 89826 by 45:
1. Determine how many times 45 goes into the leading digits of 89826.
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: 89826 / 45 = 1996 with a remainder of 6.
Proof: (1996 * 45)... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 4, 10, 9, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 4 + 10 + 9 + 15) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, -5.4, 0.6, -0.4, 5.6]
Squared Differences: [0.16, 29.16, 0.36, 0.16, 31.36]
... |
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) |
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) |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 14^2 = c^2
2. Calculate the squares: 144 + 196 = c^2
3. Sum the squares: 340 = c^2
4. Take the square root of both sides: c = √340
Final A... |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^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: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Add the fractions 5/14 and 3/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 5/14 by multiplying top and bottom by 1: 5/14.
3. Convert 3/14 by multiplying top and bottom by 1: 3/14.
4. Add the numerators: 5 + 3 = 8.
5. The resulting fraction is 8/14.
6. Simplify by divid... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 19^2 = c^2
2. Calculate the squares: 49 + 361 = c^2
3. Sum the squares: 410 = c^2
4. Take the square root of both sides: c = √410
Final Ans... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 1, 3, 7, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 1 + 3 + 7 + 10) / 5 = 27 / 5 = 5.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, -4.4, -2.4, 1.6, 4.6]
Squared Differences: [0.36, 19.36, 5.76, 2.56, 21.16]
Su... |
Add the fractions 10/4 and 11/13. 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 10/4 by multiplying top and bottom by 13: 130/52.
3. Convert 11/13 by multiplying top and bottom by 4: 44/52.
4. Add the numerators: 130 + 44 = 174.
5. The resulting fraction is 174/52.
6. Simpli... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 9^2 = c^2
2. Calculate the squares: 144 + 81 = c^2
3. Sum the squares: 225 = c^2
4. Take the square root of both sides: c = √225
Final Ans... |
Compute the quotient and remainder of 40851 divided by 18. Show the long division steps. | To divide 40851 by 18:
1. Determine how many times 18 goes into the leading digits of 40851.
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: 40851 / 18 = 2269 with a remainder of 9.
Proof: (2269 * 18)... |
Compute the quotient and remainder of 82981 divided by 68. Show the long division steps. | To divide 82981 by 68:
1. Determine how many times 68 goes into the leading digits of 82981.
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: 82981 / 68 = 1220 with a remainder of 21.
Proof: (1220 * 68... |
Multiply the following 2x2 matrices:
A =
[0, 1]
[0, -2]
B =
[-3, -2]
[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) = (0*-3) + (1*4) = 4
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (1*1) = 1
C21 = (A21 * B11) + (A22 * B21) = (0*-3) + (-2*4) = -8
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (-2*1) = -2
Result Ma... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 5, 8, 12, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 5 + 8 + 12 + 17) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.2, -6.8, -3.8, 0.2, 5.2]
Squared Differences: [27.04, 46.24, 14.44, 0.04, 27.04]
... |
Compute the quotient and remainder of 83453 divided by 32. Show the long division steps. | To divide 83453 by 32:
1. Determine how many times 32 goes into the leading digits of 83453.
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: 83453 / 32 = 2607 with a remainder of 29.
Proof: (2607 * 32... |
Compute the quotient and remainder of 45960 divided by 18. Show the long division steps. | To divide 45960 by 18:
1. Determine how many times 18 goes into the leading digits of 45960.
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: 45960 / 18 = 2553 with a remainder of 6.
Proof: (2553 * 18)... |
Compute the quotient and remainder of 67192 divided by 47. Show the long division steps. | To divide 67192 by 47:
1. Determine how many times 47 goes into the leading digits of 67192.
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: 67192 / 47 = 1429 with a remainder of 29.
Proof: (1429 * 47... |
Compute the quotient and remainder of 80875 divided by 70. Show the long division steps. | To divide 80875 by 70:
1. Determine how many times 70 goes into the leading digits of 80875.
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: 80875 / 70 = 1155 with a remainder of 25.
Proof: (1155 * 70... |
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 +... |
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 |
Multiply the following 2x2 matrices:
A =
[3, -4]
[5, -3]
B =
[-3, 4]
[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) = (3*-3) + (-4*0) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (-4*-5) = 32
C21 = (A21 * B11) + (A22 * B21) = (5*-3) + (-3*0) = -15
C22 = (A21 * B12) + (A22 * B22) = (5*4) + (-3*-5) = 35
Resu... |
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 |
Compute the quotient and remainder of 8165 divided by 8. Show the long division steps. | To divide 8165 by 8:
1. Determine how many times 8 goes into the leading digits of 8165.
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: 8165 / 8 = 1020 with a remainder of 5.
Proof: (1020 * 8) + 5 = ... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 13^2 = c^2
2. Calculate the squares: 361 + 169 = c^2
3. Sum the squares: 530 = c^2
4. Take the square root of both sides: c = √530
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 3, 9, 6, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 3 + 9 + 6 + 16) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, -4.4, 1.6, -1.4, 8.6]
Squared Differences: [19.36, 19.36, 2.56, 1.96, 73.96]
... |
Compute the quotient and remainder of 28205 divided by 62. Show the long division steps. | To divide 28205 by 62:
1. Determine how many times 62 goes into the leading digits of 28205.
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: 28205 / 62 = 454 with a remainder of 57.
Proof: (454 * 62) ... |
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) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Multiply the following 2x2 matrices:
A =
[-2, 3]
[1, -1]
B =
[1, 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) = (-2*1) + (3*-4) = -14
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (3*-4) = -18
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (-1*-4) = 5
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-1*-4) = 7
Resul... |
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 12/15 and 14/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 5, which is 15.
2. Convert 12/15 by multiplying top and bottom by 1: 12/15.
3. Convert 14/5 by multiplying top and bottom by 3: 42/15.
4. Add the numerators: 12 + 42 = 54.
5. The resulting fraction is 54/15.
6. Simplify by... |
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 +... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
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 ... |
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 +... |
Add the fractions 6/13 and 5/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 4, which is 52.
2. Convert 6/13 by multiplying top and bottom by 4: 24/52.
3. Convert 5/4 by multiplying top and bottom by 13: 65/52.
4. Add the numerators: 24 + 65 = 89.
5. The resulting fraction is 89/52.
Final Answer: 8... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 15^2 = c^2
2. Calculate the squares: 49 + 225 = c^2
3. Sum the squares: 274 = c^2
4. Take the square root of both sides: c = √274
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + 9x + -3 = 0 | 1. Identify coefficients: a=7, b=9, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -84 = 165.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √165) / 14.
5. Calculate roots: x1 = 0.2746594699046521, x2 = -1.5603737556189379.
Final ... |
Find the roots of the quadratic equation: 1x^2 + -5x + 7 = 0 | 1. Identify coefficients: a=1, b=-5, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(1)(7) = 25 - 28 = -3.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √3i) / 2.
Final Answer: x = 2.5 ± 0.8660254037844386i |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 15^2 = c^2
2. Calculate the squares: 64 + 225 = c^2
3. Sum the squares: 289 = c^2
4. Take the square root of both sides: c = √289
Final Ans... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[3, 3]
B =
[4, 5]
[-1, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*4) + (5*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (5*2) = 15
C21 = (A21 * B11) + (A22 * B21) = (3*4) + (3*-1) = 9
C22 = (A21 * B12) + (A22 * B22) = (3*5) + (3*2) = 21
Result Matri... |
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