prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 2x^2 + -8x + -6 = 0 | 1. Identify coefficients: a=2, b=-8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -48 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √112) / 4.
5. Calculate roots: x1 = 4.645751311064591, x2 = -0.6457513110645907.
Final An... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 400 + 25 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Compute the quotient and remainder of 49535 divided by 42. Show the long division steps. | To divide 49535 by 42:
1. Determine how many times 42 goes into the leading digits of 49535.
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: 49535 / 42 = 1179 with a remainder of 17.
Proof: (1179 * 42... |
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 ... |
Add the fractions 5/11 and 3/3. 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 3/3 by multiplying top and bottom by 11: 33/33.
4. Add the numerators: 15 + 33 = 48.
5. The resulting fraction is 48/33.
6. Simplify by ... |
Multiply the following 2x2 matrices:
A =
[-3, 0]
[2, 0]
B =
[5, 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) = (-3*5) + (0*-2) = -15
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (0*-4) = -3
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (0*-2) = 10
C22 = (A21 * B12) + (A22 * B22) = (2*1) + (0*-4) = 2
Result ... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 5, 6, 4, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 5 + 6 + 4 + 12) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.4, -3.6, -2.6, -4.6, 3.4]
Squared Differences: [54.76, 12.96, 6.76, 21.16, 11.56]
... |
Add the fractions 4/5 and 1/6. 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 4/5 by multiplying top and bottom by 6: 24/30.
3. Convert 1/6 by multiplying top and bottom by 5: 5/30.
4. Add the numerators: 24 + 5 = 29.
5. The resulting fraction is 29/30.
Final Answer: 29/30 |
Add the fractions 5/11 and 4/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 6, which is 66.
2. Convert 5/11 by multiplying top and bottom by 6: 30/66.
3. Convert 4/6 by multiplying top and bottom by 11: 44/66.
4. Add the numerators: 30 + 44 = 74.
5. The resulting fraction is 74/66.
6. Simplify by ... |
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) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Add the fractions 6/2 and 6/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 3, which is 6.
2. Convert 6/2 by multiplying top and bottom by 3: 18/6.
3. Convert 6/3 by multiplying top and bottom by 2: 12/6.
4. Add the numerators: 18 + 12 = 30.
5. The resulting fraction is 30/6.
6. Simplify by dividin... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 9^2 = c^2
2. Calculate the squares: 100 + 81 = c^2
3. Sum the squares: 181 = c^2
4. Take the square root of both sides: c = √181
Final Ans... |
Add the fractions 7/5 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 7/5 by multiplying top and bottom by 6: 42/30.
3. Convert 2/6 by multiplying top and bottom by 5: 10/30.
4. Add the numerators: 42 + 10 = 52.
5. The resulting fraction is 52/30.
6. Simplify by div... |
Add the fractions 14/2 and 4/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 3, which is 6.
2. Convert 14/2 by multiplying top and bottom by 3: 42/6.
3. Convert 4/3 by multiplying top and bottom by 2: 8/6.
4. Add the numerators: 42 + 8 = 50.
5. The resulting fraction is 50/6.
6. Simplify by dividing... |
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 = 11 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 11^2 + 3^2 = c^2
2. Calculate the squares: 121 + 9 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answ... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 15, 19, 4, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 15 + 19 + 4 + 10) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.0, 3.0, 7.0, -8.0, -2.0]
Squared Differences: [0.0, 9.0, 49.0, 64.0, 4.0]
Sum... |
Compute the quotient and remainder of 16773 divided by 11. Show the long division steps. | To divide 16773 by 11:
1. Determine how many times 11 goes into the leading digits of 16773.
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: 16773 / 11 = 1524 with a remainder of 9.
Proof: (1524 * 11)... |
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) |
Add the fractions 7/3 and 15/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 6, which is 6.
2. Convert 7/3 by multiplying top and bottom by 2: 14/6.
3. Convert 15/6 by multiplying top and bottom by 1: 15/6.
4. Add the numerators: 14 + 15 = 29.
5. The resulting fraction is 29/6.
Final Answer: 29/6 |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 11, 12, 13, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 11 + 12 + 13 + 16) / 5 = 72 / 5 = 14.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.6, -3.4, -2.4, -1.4, 1.6]
Squared Differences: [31.36, 11.56, 5.76, 1.96, 2.56]... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 67125 divided by 13. Show the long division steps. | To divide 67125 by 13:
1. Determine how many times 13 goes into the leading digits of 67125.
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: 67125 / 13 = 5163 with a remainder of 6.
Proof: (5163 * 13)... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 6, 12, 15, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 6 + 12 + 15 + 4) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.6, -5.4, 0.6, 3.6, -7.4]
Squared Differences: [73.96, 29.16, 0.36, 12.96, 54.76]
... |
Multiply the following 2x2 matrices:
A =
[3, 3]
[5, 4]
B =
[-3, -3]
[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) + (3*0) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (3*-5) = -24
C21 = (A21 * B11) + (A22 * B21) = (5*-3) + (4*0) = -15
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (4*-5) = -35
Resu... |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 289 + 25 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
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: 8x^2 + 0x + 4 = 0 | 1. Identify coefficients: a=8, b=0, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(8)(4) = 0 - 128 = -128.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √128i) / 16.
Final Answer: x = 0.0 ± 0.7071067811865476i |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 7^2 = c^2
2. Calculate the squares: 144 + 49 = c^2
3. Sum the squares: 193 = c^2
4. Take the square root of both sides: c = √193
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[4, 0]
B =
[-5, -5]
[4, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-5) + (4*4) = 21
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (4*-5) = -15
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (0*4) = -20
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (0*-5) = -20
Re... |
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) |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 18, 20, 11, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 18 + 20 + 11 + 14) / 5 = 77 / 5 = 15.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, 2.6, 4.6, -4.4, -1.4]
Squared Differences: [1.96, 6.76, 21.16, 19.36, 1.96]... |
Find the roots of the quadratic equation: 8x^2 + 5x + -8 = 0 | 1. Identify coefficients: a=8, b=5, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -256 = 281.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √281) / 16.
5. Calculate roots: x1 = 0.7351909133900132, x2 = -1.3601909133900132.
Final... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Add the fractions 7/5 and 3/7. 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 7/5 by multiplying top and bottom by 7: 49/35.
3. Convert 3/7 by multiplying top and bottom by 5: 15/35.
4. Add the numerators: 49 + 15 = 64.
5. The resulting fraction is 64/35.
Final Answer: 64/3... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
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 |
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) = 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 |
Add the fractions 15/3 and 14/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 4, which is 12.
2. Convert 15/3 by multiplying top and bottom by 4: 60/12.
3. Convert 14/4 by multiplying top and bottom by 3: 42/12.
4. Add the numerators: 60 + 42 = 102.
5. The resulting fraction is 102/12.
6. Simplify by... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 4. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 4^2 = c^2
2. Calculate the squares: 169 + 16 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
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 |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[-2, 5]
B =
[-4, -5]
[-3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-4) + (-2*-3) = 26
C12 = (A11 * B12) + (A12 * B22) = (-5*-5) + (-2*-4) = 33
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (5*-3) = -7
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (5*-4) = -10... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 8, 19, 15, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 8 + 19 + 15 + 13) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, -6.2, 4.8, 0.8, -1.2]
Squared Differences: [3.24, 38.44, 23.04, 0.64, 1.44]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 16, 18, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 16 + 18 + 9 + 18) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-11.4, 3.6, 5.6, -3.4, 5.6]
Squared Differences: [129.96, 12.96, 31.36, 11.56, 31.3... |
Add the fractions 9/12 and 1/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 4, which is 12.
2. Convert 9/12 by multiplying top and bottom by 1: 9/12.
3. Convert 1/4 by multiplying top and bottom by 3: 3/12.
4. Add the numerators: 9 + 3 = 12.
5. The resulting fraction is 12/12.
6. Simplify by divid... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 10x^2 + 4x + 10 = 0 | 1. Identify coefficients: a=10, b=4, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(10)(10) = 16 - 400 = -384.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √384i) / 20.
Final Answer: x = -0.2 ± 0.9797958971132712i |
Compute the quotient and remainder of 63180 divided by 4. Show the long division steps. | To divide 63180 by 4:
1. Determine how many times 4 goes into the leading digits of 63180.
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: 63180 / 4 = 15795 with a remainder of 0.
Proof: (15795 * 4) +... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[5, 1]
[-3, 4]
B =
[-3, 4]
[-3, 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) = (5*-3) + (1*-3) = -18
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (1*0) = 20
C21 = (A21 * B11) + (A22 * B21) = (-3*-3) + (4*-3) = -3
C22 = (A21 * B12) + (A22 * B22) = (-3*4) + (4*0) = -12
Resul... |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
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) |
Compute the quotient and remainder of 18208 divided by 5. Show the long division steps. | To divide 18208 by 5:
1. Determine how many times 5 goes into the leading digits of 18208.
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: 18208 / 5 = 3641 with a remainder of 3.
Proof: (3641 * 5) + 3... |
Add the fractions 3/14 and 15/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 5, which is 70.
2. Convert 3/14 by multiplying top and bottom by 5: 15/70.
3. Convert 15/5 by multiplying top and bottom by 14: 210/70.
4. Add the numerators: 15 + 210 = 225.
5. The resulting fraction is 225/70.
6. Simplif... |
Add the fractions 14/7 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 14/7 by multiplying top and bottom by 10: 140/70.
3. Convert 8/10 by multiplying top and bottom by 7: 56/70.
4. Add the numerators: 140 + 56 = 196.
5. The resulting fraction is 196/70.
6. Simplif... |
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 +... |
Find the roots of the quadratic equation: 7x^2 + -2x + -6 = 0 | 1. Identify coefficients: a=7, b=-2, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -168 = 172.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √172) / 14.
5. Calculate roots: x1 = 1.0796340749002857, x2 = -0.793919789186.
Final Answ... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
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 = 0.6104686356149273, x2 = -1.3104686356149273.
Fina... |
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, 16, 11, 8, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 16 + 11 + 8 + 8) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, 4.4, -0.6, -3.6, -3.6]
Squared Differences: [11.56, 19.36, 0.36, 12.96, 12.96]... |
Multiply the following 2x2 matrices:
A =
[1, 1]
[-1, 3]
B =
[-2, -1]
[3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-2) + (1*3) = 1
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (1*-2) = -3
C21 = (A21 * B11) + (A22 * B21) = (-1*-2) + (3*3) = 11
C22 = (A21 * B12) + (A22 * B22) = (-1*-1) + (3*-2) = -5
Result... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 17^2 = c^2
2. Calculate the squares: 225 + 289 = c^2
3. Sum the squares: 514 = c^2
4. Take the square root of both sides: c = √514
Final A... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[0, 1]
B =
[1, -2]
[-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) = (0*1) + (-4*-4) = 16
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (-4*-2) = 8
C21 = (A21 * B11) + (A22 * B21) = (0*1) + (1*-4) = -4
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (1*-2) = -2
Result... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Compute the quotient and remainder of 35504 divided by 83. Show the long division steps. | To divide 35504 by 83:
1. Determine how many times 83 goes into the leading digits of 35504.
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: 35504 / 83 = 427 with a remainder of 63.
Proof: (427 * 83) ... |
Add the fractions 11/14 and 1/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 13, which is 182.
2. Convert 11/14 by multiplying top and bottom by 13: 143/182.
3. Convert 1/13 by multiplying top and bottom by 14: 14/182.
4. Add the numerators: 143 + 14 = 157.
5. The resulting fraction is 157/182.
Fin... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[2, -2]
B =
[-1, -5]
[5, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-1) + (4*5) = 23
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (4*0) = 15
C21 = (A21 * B11) + (A22 * B21) = (2*-1) + (-2*5) = -12
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-2*0) = -10
Res... |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 46175 divided by 9. Show the long division steps. | To divide 46175 by 9:
1. Determine how many times 9 goes into the leading digits of 46175.
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: 46175 / 9 = 5130 with a remainder of 5.
Proof: (5130 * 9) + 5... |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 196 + 25 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final Ans... |
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: 10x^2 + 0x + -6 = 0 | 1. Identify coefficients: a=10, b=0, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -240 = 240.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √240) / 20.
5. Calculate roots: x1 = 0.7745966692414834, x2 = -0.7745966692414834.
Final ... |
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) |
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: 3x^2 + -1x + -9 = 0 | 1. Identify coefficients: a=3, b=-1, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -108 = 109.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √109) / 6.
5. Calculate roots: x1 = 1.9067177514850917, x2 = -1.5733844181517584.
Final A... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (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 ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 14^2 = c^2
2. Calculate the squares: 361 + 196 = c^2
3. Sum the squares: 557 = c^2
4. Take the square root of both sides: c = √557
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 8^2 = c^2
2. Calculate the squares: 121 + 64 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Add the fractions 5/6 and 8/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 10, which is 30.
2. Convert 5/6 by multiplying top and bottom by 5: 25/30.
3. Convert 8/10 by multiplying top and bottom by 3: 24/30.
4. Add the numerators: 25 + 24 = 49.
5. The resulting fraction is 49/30.
Final Answer: 49... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 19^2 = c^2
2. Calculate the squares: 9 + 361 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
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) |
Find the roots of the quadratic equation: 7x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=7, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -28 = 92.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √92) / 14.
5. Calculate roots: x1 = 1.2565473604732456, x2 = -0.11369021761610273.
Final A... |
Find the roots of the quadratic equation: 3x^2 + 3x + 7 = 0 | 1. Identify coefficients: a=3, b=3, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(3)(7) = 9 - 84 = -75.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √75i) / 6.
Final Answer: x = -0.5 ± 1.4433756729740645i |
Compute the quotient and remainder of 59477 divided by 23. Show the long division steps. | To divide 59477 by 23:
1. Determine how many times 23 goes into the leading digits of 59477.
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: 59477 / 23 = 2585 with a remainder of 22.
Proof: (2585 * 23... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
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 +... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.