prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
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) |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 16. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 16^2 = c^2
2. Calculate the squares: 225 + 256 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final A... |
Compute the quotient and remainder of 39981 divided by 46. Show the long division steps. | To divide 39981 by 46:
1. Determine how many times 46 goes into the leading digits of 39981.
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: 39981 / 46 = 869 with a remainder of 7.
Proof: (869 * 46) +... |
Multiply the following 2x2 matrices:
A =
[2, -1]
[2, 2]
B =
[0, -2]
[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) = (2*0) + (-1*5) = -5
C12 = (A11 * B12) + (A12 * B22) = (2*-2) + (-1*0) = -4
C21 = (A21 * B11) + (A22 * B21) = (2*0) + (2*5) = 10
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (2*0) = -4
Result Ma... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
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 quotient and remainder of 57893 divided by 12. Show the long division steps. | To divide 57893 by 12:
1. Determine how many times 12 goes into the leading digits of 57893.
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: 57893 / 12 = 4824 with a remainder of 5.
Proof: (4824 * 12)... |
Find the roots of the quadratic equation: 10x^2 + -3x + 9 = 0 | 1. Identify coefficients: a=10, b=-3, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(10)(9) = 9 - 360 = -351.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √351i) / 20.
Final Answer: x = 0.15 ± 0.9367496997597596i |
Multiply the following 2x2 matrices:
A =
[2, 1]
[1, -1]
B =
[4, -4]
[0, 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) = (2*4) + (1*0) = 8
C12 = (A11 * B12) + (A12 * B22) = (2*-4) + (1*2) = -6
C21 = (A21 * B11) + (A22 * B21) = (1*4) + (-1*0) = 4
C22 = (A21 * B12) + (A22 * B22) = (1*-4) + (-1*2) = -6
Result Matr... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 10, 16, 6, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 10 + 16 + 6 + 7) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, 1.4, 7.4, -2.6, -1.6]
Squared Differences: [21.16, 1.96, 54.76, 6.76, 2.56]
... |
Compute the quotient and remainder of 47962 divided by 27. Show the long division steps. | To divide 47962 by 27:
1. Determine how many times 27 goes into the leading digits of 47962.
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: 47962 / 27 = 1776 with a remainder of 10.
Proof: (1776 * 27... |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Compute the quotient and remainder of 53376 divided by 99. Show the long division steps. | To divide 53376 by 99:
1. Determine how many times 99 goes into the leading digits of 53376.
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: 53376 / 99 = 539 with a remainder of 15.
Proof: (539 * 99) ... |
Find the roots of the quadratic equation: 9x^2 + -6x + -9 = 0 | 1. Identify coefficients: a=9, b=-6, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -324 = 360.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √360) / 18.
5. Calculate roots: x1 = 1.387425886722793, x2 = -0.7207592200561265.
Final ... |
Multiply the following 2x2 matrices:
A =
[3, 3]
[5, -2]
B =
[-4, -5]
[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) = (3*-4) + (3*4) = 0
C12 = (A11 * B12) + (A12 * B22) = (3*-5) + (3*2) = -9
C21 = (A21 * B11) + (A22 * B21) = (5*-4) + (-2*4) = -28
C22 = (A21 * B12) + (A22 * B22) = (5*-5) + (-2*2) = -29
Result... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 12^2 = c^2
2. Calculate the squares: 256 + 144 = c^2
3. Sum the squares: 400 = c^2
4. Take the square root of both sides: c = √400
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 9, 16, 14, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 9 + 16 + 14 + 16) / 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, -5.2, 1.8, -0.2, 1.8]
Squared Differences: [3.24, 27.04, 3.24, 0.04, 3.24]
... |
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: [4, 17, 14, 8, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 17 + 14 + 8 + 9) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 6.6, 3.6, -2.4, -1.4]
Squared Differences: [40.96, 43.56, 12.96, 5.76, 1.96]
... |
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 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 ... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[3, 1]
B =
[2, 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) = (-1*2) + (5*-2) = -12
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (5*-4) = -22
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (1*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (1*-4) = 2
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 18^2 = c^2
2. Calculate the squares: 81 + 324 = c^2
3. Sum the squares: 405 = c^2
4. Take the square root of both sides: c = √405
Final Ans... |
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 |
Add the fractions 1/3 and 15/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 10, which is 30.
2. Convert 1/3 by multiplying top and bottom by 10: 10/30.
3. Convert 15/10 by multiplying top and bottom by 3: 45/30.
4. Add the numerators: 10 + 45 = 55.
5. The resulting fraction is 55/30.
6. Simplify by... |
Multiply the following 2x2 matrices:
A =
[4, -5]
[0, -3]
B =
[2, -5]
[-4, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*2) + (-5*-4) = 28
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (-5*-4) = 0
C21 = (A21 * B11) + (A22 * B21) = (0*2) + (-3*-4) = 12
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (-3*-4) = 12
Resu... |
Multiply the following 2x2 matrices:
A =
[3, 1]
[2, 1]
B =
[0, 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) = (3*0) + (1*2) = 2
C12 = (A11 * B12) + (A12 * B22) = (3*2) + (1*-4) = 2
C21 = (A21 * B11) + (A22 * B21) = (2*0) + (1*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (1*-4) = 0
Result Matrix C... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 17, 11, 10, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 17 + 11 + 10 + 9) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, 5.2, -0.8, -1.8, -2.8]
Squared Differences: [0.04, 27.04, 0.64, 3.24, 7.84]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 16, 5, 7, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 16 + 5 + 7 + 6) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, 8.8, -2.2, -0.2, -1.2]
Squared Differences: [27.04, 77.44, 4.84, 0.04, 1.44]
... |
Add the fractions 3/14 and 12/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 4, which is 28.
2. Convert 3/14 by multiplying top and bottom by 2: 6/28.
3. Convert 12/4 by multiplying top and bottom by 7: 84/28.
4. Add the numerators: 6 + 84 = 90.
5. The resulting fraction is 90/28.
6. Simplify by di... |
Find the roots of the quadratic equation: 8x^2 + 8x + 10 = 0 | 1. Identify coefficients: a=8, b=8, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(8)(10) = 64 - 320 = -256.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √256i) / 16.
Final Answer: x = -0.5 ± 1.0i |
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 |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 20^2 = c^2
2. Calculate the squares: 49 + 400 = c^2
3. Sum the squares: 449 = c^2
4. Take the square root of both sides: c = √449
Final Ans... |
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 indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^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: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
Compute the quotient and remainder of 18460 divided by 83. Show the long division steps. | To divide 18460 by 83:
1. Determine how many times 83 goes into the leading digits of 18460.
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: 18460 / 83 = 222 with a remainder of 34.
Proof: (222 * 83) ... |
Find the roots of the quadratic equation: 2x^2 + 3x + -9 = 0 | 1. Identify coefficients: a=2, b=3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -72 = 81.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √81) / 4.
5. Calculate roots: x1 = 1.5, x2 = -3.0.
Final Answer: x = 1.5, -3.0 |
Add the fractions 2/14 and 12/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 2/14 by multiplying top and bottom by 3: 6/42.
3. Convert 12/3 by multiplying top and bottom by 14: 168/42.
4. Add the numerators: 6 + 168 = 174.
5. The resulting fraction is 174/42.
6. Simplify ... |
Multiply the following 2x2 matrices:
A =
[5, -1]
[5, 1]
B =
[4, -3]
[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) + (-1*3) = 17
C12 = (A11 * B12) + (A12 * B22) = (5*-3) + (-1*4) = -19
C21 = (A21 * B11) + (A22 * B21) = (5*4) + (1*3) = 23
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (1*4) = -11
Result ... |
Add the fractions 3/5 and 5/7. 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 3/5 by multiplying top and bottom by 7: 21/35.
3. Convert 5/7 by multiplying top and bottom by 5: 25/35.
4. Add the numerators: 21 + 25 = 46.
5. The resulting fraction is 46/35.
Final Answer: 46/3... |
Add the fractions 12/12 and 3/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 7, which is 84.
2. Convert 12/12 by multiplying top and bottom by 7: 84/84.
3. Convert 3/7 by multiplying top and bottom by 12: 36/84.
4. Add the numerators: 84 + 36 = 120.
5. The resulting fraction is 120/84.
6. Simplify ... |
Compute the quotient and remainder of 13481 divided by 5. Show the long division steps. | To divide 13481 by 5:
1. Determine how many times 5 goes into the leading digits of 13481.
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: 13481 / 5 = 2696 with a remainder of 1.
Proof: (2696 * 5) + 1... |
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 = 12 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: 12^2 + 18^2 = c^2
2. Calculate the squares: 144 + 324 = c^2
3. Sum the squares: 468 = c^2
4. Take the square root of both sides: c = √468
Final A... |
Multiply the following 2x2 matrices:
A =
[1, 2]
[1, 4]
B =
[-5, 2]
[0, 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) = (1*-5) + (2*0) = -5
C12 = (A11 * B12) + (A12 * B22) = (1*2) + (2*1) = 4
C21 = (A21 * B11) + (A22 * B21) = (1*-5) + (4*0) = -5
C22 = (A21 * B12) + (A22 * B22) = (1*2) + (4*1) = 6
Result Matrix... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 5, 7, 11, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 5 + 7 + 11 + 16) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.2, -5.8, -3.8, 0.2, 5.2]
Squared Differences: [17.64, 33.64, 14.44, 0.04, 27.04]
... |
Add the fractions 7/11 and 14/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 15, which is 165.
2. Convert 7/11 by multiplying top and bottom by 15: 105/165.
3. Convert 14/15 by multiplying top and bottom by 11: 154/165.
4. Add the numerators: 105 + 154 = 259.
5. The resulting fraction is 259/165.
F... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 9^2 = c^2
2. Calculate the squares: 36 + 81 = c^2
3. Sum the squares: 117 = c^2
4. Take the square root of both sides: c = √117
Final Answe... |
Add the fractions 3/4 and 7/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 4, which is 4.
2. Convert 3/4 by multiplying top and bottom by 1: 3/4.
3. Convert 7/4 by multiplying top and bottom by 1: 7/4.
4. Add the numerators: 3 + 7 = 10.
5. The resulting fraction is 10/4.
6. Simplify by dividing nu... |
Add the fractions 15/12 and 2/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 13, which is 156.
2. Convert 15/12 by multiplying top and bottom by 13: 195/156.
3. Convert 2/13 by multiplying top and bottom by 12: 24/156.
4. Add the numerators: 195 + 24 = 219.
5. The resulting fraction is 219/156.
6. ... |
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) |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 12, 2, 6, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 12 + 2 + 6 + 12) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.6, 3.6, -6.4, -2.4, 3.6]
Squared Differences: [2.56, 12.96, 40.96, 5.76, 12.96]
... |
Compute the quotient and remainder of 27154 divided by 5. Show the long division steps. | To divide 27154 by 5:
1. Determine how many times 5 goes into the leading digits of 27154.
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: 27154 / 5 = 5430 with a remainder of 4.
Proof: (5430 * 5) + 4... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 79143 divided by 30. Show the long division steps. | To divide 79143 by 30:
1. Determine how many times 30 goes into the leading digits of 79143.
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: 79143 / 30 = 2638 with a remainder of 3.
Proof: (2638 * 30)... |
Compute the quotient and remainder of 60879 divided by 98. Show the long division steps. | To divide 60879 by 98:
1. Determine how many times 98 goes into the leading digits of 60879.
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: 60879 / 98 = 621 with a remainder of 21.
Proof: (621 * 98) ... |
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 =
[-5, 2]
[5, 4]
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) = (-5*5) + (2*-3) = -31
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (2*5) = 5
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (4*-3) = 13
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (4*5) = 25
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[2, 0]
B =
[-1, -5]
[-1, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-1) + (-1*-1) = 5
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (-1*-3) = 23
C21 = (A21 * B11) + (A22 * B21) = (2*-1) + (0*-1) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (0*-3) = -10
R... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[3, -1]
[2, -5]
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) = (3*-5) + (-1*3) = -18
C12 = (A11 * B12) + (A12 * B22) = (3*3) + (-1*1) = 8
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (-5*3) = -25
C22 = (A21 * B12) + (A22 * B22) = (2*3) + (-5*1) = 1
Result ... |
Multiply the following 2x2 matrices:
A =
[4, -1]
[1, 0]
B =
[-5, 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) = (4*-5) + (-1*-5) = -15
C12 = (A11 * B12) + (A12 * B22) = (4*1) + (-1*5) = -1
C21 = (A21 * B11) + (A22 * B21) = (1*-5) + (0*-5) = -5
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (0*5) = 1
Result ... |
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 |
Add the fractions 3/11 and 1/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 5, which is 55.
2. Convert 3/11 by multiplying top and bottom by 5: 15/55.
3. Convert 1/5 by multiplying top and bottom by 11: 11/55.
4. Add the numerators: 15 + 11 = 26.
5. The resulting fraction is 26/55.
Final Answer: 2... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 10^2 = c^2
2. Calculate the squares: 49 + 100 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 19, 18, 11, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 19 + 18 + 11 + 4) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 7.8, 6.8, -0.2, -7.2]
Squared Differences: [51.84, 60.84, 46.24, 0.04, 51.84]... |
Add the fractions 15/6 and 14/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 15, which is 30.
2. Convert 15/6 by multiplying top and bottom by 5: 75/30.
3. Convert 14/15 by multiplying top and bottom by 2: 28/30.
4. Add the numerators: 75 + 28 = 103.
5. The resulting fraction is 103/30.
Final Answer... |
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 ... |
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 |
Multiply the following 2x2 matrices:
A =
[1, -4]
[2, -2]
B =
[-4, -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*-4) + (-4*-1) = 0
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (-4*-4) = 11
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (-2*-1) = -6
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-2*-4) = -2
Re... |
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 +... |
Multiply the following 2x2 matrices:
A =
[2, -1]
[-4, 0]
B =
[1, -5]
[-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) = (2*1) + (-1*-5) = 7
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-1*-5) = -5
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (0*-5) = -4
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (0*-5) = 20
Resu... |
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 quotient and remainder of 89879 divided by 73. Show the long division steps. | To divide 89879 by 73:
1. Determine how many times 73 goes into the leading digits of 89879.
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: 89879 / 73 = 1231 with a remainder of 16.
Proof: (1231 * 73... |
Find the roots of the quadratic equation: 9x^2 + -1x + -5 = 0 | 1. Identify coefficients: a=9, b=-1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -180 = 181.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √181) / 18.
5. Calculate roots: x1 = 0.8029791137263173, x2 = -0.6918680026152061.
Final ... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^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: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
Compute the quotient and remainder of 74067 divided by 73. Show the long division steps. | To divide 74067 by 73:
1. Determine how many times 73 goes into the leading digits of 74067.
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: 74067 / 73 = 1014 with a remainder of 45.
Proof: (1014 * 73... |
Add the fractions 2/12 and 8/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 6, which is 12.
2. Convert 2/12 by multiplying top and bottom by 1: 2/12.
3. Convert 8/6 by multiplying top and bottom by 2: 16/12.
4. Add the numerators: 2 + 16 = 18.
5. The resulting fraction is 18/12.
6. Simplify by div... |
Add the fractions 5/8 and 5/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 9, which is 72.
2. Convert 5/8 by multiplying top and bottom by 9: 45/72.
3. Convert 5/9 by multiplying top and bottom by 8: 40/72.
4. Add the numerators: 45 + 40 = 85.
5. The resulting fraction is 85/72.
Final Answer: 85/7... |
Add the fractions 5/7 and 8/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 8, which is 56.
2. Convert 5/7 by multiplying top and bottom by 8: 40/56.
3. Convert 8/8 by multiplying top and bottom by 7: 56/56.
4. Add the numerators: 40 + 56 = 96.
5. The resulting fraction is 96/56.
6. Simplify by div... |
Multiply the following 2x2 matrices:
A =
[0, 5]
[0, 3]
B =
[5, 5]
[-5, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*5) + (5*-5) = -25
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (5*2) = 10
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (3*-5) = -15
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (3*2) = 6
Result Mat... |
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) |
Find the roots of the quadratic equation: 8x^2 + 3x + 4 = 0 | 1. Identify coefficients: a=8, b=3, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(8)(4) = 9 - 128 = -119.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √119i) / 16.
Final Answer: x = -0.1875 ± 0.6817945071647321i |
Compute the quotient and remainder of 50209 divided by 58. Show the long division steps. | To divide 50209 by 58:
1. Determine how many times 58 goes into the leading digits of 50209.
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: 50209 / 58 = 865 with a remainder of 39.
Proof: (865 * 58) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 7, 14, 8, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 7 + 14 + 8 + 16) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, -4.0, 3.0, -3.0, 5.0]
Squared Differences: [1.0, 16.0, 9.0, 9.0, 25.0]
Sum... |
Add the fractions 5/10 and 2/14. 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 5/10 by multiplying top and bottom by 7: 35/70.
3. Convert 2/14 by multiplying top and bottom by 5: 10/70.
4. Add the numerators: 35 + 10 = 45.
5. The resulting fraction is 45/70.
6. Simplify by... |
Add the fractions 10/6 and 5/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 9, which is 18.
2. Convert 10/6 by multiplying top and bottom by 3: 30/18.
3. Convert 5/9 by multiplying top and bottom by 2: 10/18.
4. Add the numerators: 30 + 10 = 40.
5. The resulting fraction is 40/18.
6. Simplify by di... |
Add the fractions 14/5 and 4/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 11, which is 55.
2. Convert 14/5 by multiplying top and bottom by 11: 154/55.
3. Convert 4/11 by multiplying top and bottom by 5: 20/55.
4. Add the numerators: 154 + 20 = 174.
5. The resulting fraction is 174/55.
Final Answ... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
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) |
Compute the indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^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: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
Compute the quotient and remainder of 12970 divided by 90. Show the long division steps. | To divide 12970 by 90:
1. Determine how many times 90 goes into the leading digits of 12970.
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: 12970 / 90 = 144 with a remainder of 10.
Proof: (144 * 90) ... |
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) |
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 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) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.