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Compute the quotient and remainder of 54164 divided by 93. Show the long division steps. | To divide 54164 by 93:
1. Determine how many times 93 goes into the leading digits of 54164.
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: 54164 / 93 = 582 with a remainder of 38.
Proof: (582 * 93) ... |
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 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) |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 10^2 = c^2
2. Calculate the squares: 225 + 100 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final A... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the roots of the quadratic equation: 10x^2 + 8x + -5 = 0 | 1. Identify coefficients: a=10, b=8, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -200 = 264.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √264) / 20.
5. Calculate roots: x1 = 0.41240384046359607, x2 = -1.2124038404635962.
Fin... |
Find the roots of the quadratic equation: 2x^2 + -10x + 8 = 0 | 1. Identify coefficients: a=2, b=-10, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 64 = 36.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √36) / 4.
5. Calculate roots: x1 = 4.0, x2 = 1.0.
Final Answer: x = 4.0, 1.0 |
Add the fractions 7/8 and 15/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 10, which is 40.
2. Convert 7/8 by multiplying top and bottom by 5: 35/40.
3. Convert 15/10 by multiplying top and bottom by 4: 60/40.
4. Add the numerators: 35 + 60 = 95.
5. The resulting fraction is 95/40.
6. Simplify by ... |
Find the roots of the quadratic equation: 5x^2 + -5x + -7 = 0 | 1. Identify coefficients: a=5, b=-5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -140 = 165.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √165) / 10.
5. Calculate roots: x1 = 1.784523257866513, x2 = -0.7845232578665129.
Final ... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Compute the quotient and remainder of 68738 divided by 47. Show the long division steps. | To divide 68738 by 47:
1. Determine how many times 47 goes into the leading digits of 68738.
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: 68738 / 47 = 1462 with a remainder of 24.
Proof: (1462 * 47... |
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 |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[0, 2]
B =
[-5, -2]
[3, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-5) + (0*3) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*-2) + (0*5) = 4
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (2*3) = 6
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (2*5) = 10
Result Ma... |
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 |
Add the fractions 13/5 and 15/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 8, which is 40.
2. Convert 13/5 by multiplying top and bottom by 8: 104/40.
3. Convert 15/8 by multiplying top and bottom by 5: 75/40.
4. Add the numerators: 104 + 75 = 179.
5. The resulting fraction is 179/40.
Final Answer... |
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_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 |
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 |
Find the roots of the quadratic equation: 10x^2 + 1x + -4 = 0 | 1. Identify coefficients: a=10, b=1, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -160 = 161.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √161) / 20.
5. Calculate roots: x1 = 0.584428877022476, x2 = -0.684428877022476.
Final A... |
Find the roots of the quadratic equation: 2x^2 + 4x + 4 = 0 | 1. Identify coefficients: a=2, b=4, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(2)(4) = 16 - 32 = -16.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √16i) / 4.
Final Answer: x = -1.0 ± 1.0i |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 6x^2 + -7x + -10 = 0 | 1. Identify coefficients: a=6, b=-7, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -240 = 289.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √289) / 12.
5. Calculate roots: x1 = 2.0, x2 = -0.8333333333333334.
Final Answer: x = 2... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 6^2 = c^2
2. Calculate the squares: 196 + 36 = c^2
3. Sum the squares: 232 = c^2
4. Take the square root of both sides: c = √232
Final Ans... |
Compute the indefinite integral of f(x) = 3x^3. | To find the indefinite integral ∫ 3x^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: 3 / 4 = 3/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^3 dx = 3/4x^4 + C |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Add the fractions 10/8 and 13/8. 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 10/8 by multiplying top and bottom by 1: 10/8.
3. Convert 13/8 by multiplying top and bottom by 1: 13/8.
4. Add the numerators: 10 + 13 = 23.
5. The resulting fraction is 23/8.
Final Answer: 23/8 |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 7, 14, 12, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 7 + 14 + 12 + 5) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.0, -4.0, 3.0, 1.0, -6.0]
Squared Differences: [36.0, 16.0, 9.0, 1.0, 36.0]
Sum... |
Add the fractions 7/10 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 9, which is 90.
2. Convert 7/10 by multiplying top and bottom by 9: 63/90.
3. Convert 15/9 by multiplying top and bottom by 10: 150/90.
4. Add the numerators: 63 + 150 = 213.
5. The resulting fraction is 213/90.
6. Simplif... |
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 |
Compute the quotient and remainder of 30622 divided by 38. Show the long division steps. | To divide 30622 by 38:
1. Determine how many times 38 goes into the leading digits of 30622.
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: 30622 / 38 = 805 with a remainder of 32.
Proof: (805 * 38) ... |
Multiply the following 2x2 matrices:
A =
[3, 1]
[-3, 3]
B =
[-1, -3]
[-2, -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*-1) + (1*-2) = -5
C12 = (A11 * B12) + (A12 * B22) = (3*-3) + (1*-2) = -11
C21 = (A21 * B11) + (A22 * B21) = (-3*-1) + (3*-2) = -3
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (3*-2) = 3
Res... |
Multiply the following 2x2 matrices:
A =
[2, -5]
[1, -2]
B =
[5, 0]
[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) = (2*5) + (-5*3) = -5
C12 = (A11 * B12) + (A12 * B22) = (2*0) + (-5*-4) = 20
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-2*3) = -1
C22 = (A21 * B12) + (A22 * B22) = (1*0) + (-2*-4) = 8
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 4^2 = c^2
2. Calculate the squares: 64 + 16 = c^2
3. Sum the squares: 80 = c^2
4. Take the square root of both sides: c = √80
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[3, 4]
[-2, 5]
B =
[4, -4]
[-1, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*4) + (4*-1) = 8
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (4*5) = 8
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (5*-1) = -13
C22 = (A21 * B12) + (A22 * B22) = (-2*-4) + (5*5) = 33
Result M... |
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 |
Compute the quotient and remainder of 41644 divided by 89. Show the long division steps. | To divide 41644 by 89:
1. Determine how many times 89 goes into the leading digits of 41644.
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: 41644 / 89 = 467 with a remainder of 81.
Proof: (467 * 89) ... |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 3^2 + 13^2 = c^2
2. Calculate the squares: 9 + 169 = c^2
3. Sum the squares: 178 = c^2
4. Take the square root of both sides: c = √178
Final Answ... |
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 |
Find the roots of the quadratic equation: 3x^2 + -8x + 10 = 0 | 1. Identify coefficients: a=3, b=-8, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(3)(10) = 64 - 120 = -56.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √56i) / 6.
Final Answer: x = 1.3333333333333333 ± 1.247219128924647i |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 12^2 = c^2
2. Calculate the squares: 100 + 144 = c^2
3. Sum the squares: 244 = c^2
4. Take the square root of both sides: c = √244
Final A... |
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... |
Add the fractions 13/15 and 8/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 11, which is 165.
2. Convert 13/15 by multiplying top and bottom by 11: 143/165.
3. Convert 8/11 by multiplying top and bottom by 15: 120/165.
4. Add the numerators: 143 + 120 = 263.
5. The resulting fraction is 263/165.
F... |
Compute the quotient and remainder of 65422 divided by 27. Show the long division steps. | To divide 65422 by 27:
1. Determine how many times 27 goes into the leading digits of 65422.
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: 65422 / 27 = 2423 with a remainder of 1.
Proof: (2423 * 27)... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 11, 16, 15, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 11 + 16 + 15 + 20) / 5 = 76 / 5 = 15.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, -4.2, 0.8, -0.2, 4.8]
Squared Differences: [1.44, 17.64, 0.64, 0.04, 23.04]... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[-1, -1]
B =
[5, 0]
[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) + (4*2) = -7
C12 = (A11 * B12) + (A12 * B22) = (-3*0) + (4*-4) = -16
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (-1*2) = -7
C22 = (A21 * B12) + (A22 * B22) = (-1*0) + (-1*-4) = 4
Resul... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[-4, -3]
B =
[-2, 1]
[-3, -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) = (-1*-2) + (-5*-3) = 17
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (-5*-3) = 14
C21 = (A21 * B11) + (A22 * B21) = (-4*-2) + (-3*-3) = 17
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (-3*-3) = 5
... |
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 |
Compute the quotient and remainder of 68146 divided by 57. Show the long division steps. | To divide 68146 by 57:
1. Determine how many times 57 goes into the leading digits of 68146.
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: 68146 / 57 = 1195 with a remainder of 31.
Proof: (1195 * 57... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 4^2 = c^2
2. Calculate the squares: 289 + 16 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
Compute the quotient and remainder of 65030 divided by 41. Show the long division steps. | To divide 65030 by 41:
1. Determine how many times 41 goes into the leading digits of 65030.
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: 65030 / 41 = 1586 with a remainder of 4.
Proof: (1586 * 41)... |
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 +... |
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... |
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 |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 7, 4, 18, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 7 + 4 + 18 + 19) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -4.2, -7.2, 6.8, 7.8]
Squared Differences: [10.24, 17.64, 51.84, 46.24, 60.84]... |
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_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 |
Add the fractions 6/15 and 11/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 14, which is 210.
2. Convert 6/15 by multiplying top and bottom by 14: 84/210.
3. Convert 11/14 by multiplying top and bottom by 15: 165/210.
4. Add the numerators: 84 + 165 = 249.
5. The resulting fraction is 249/210.
6. ... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Multiply the following 2x2 matrices:
A =
[5, 1]
[-5, 5]
B =
[-5, 4]
[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) = (5*-5) + (1*1) = -24
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (1*-4) = 16
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (5*1) = 30
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (5*-4) = -40
Resul... |
Find the roots of the quadratic equation: 8x^2 + -3x + -9 = 0 | 1. Identify coefficients: a=8, b=-3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -288 = 297.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √297) / 16.
5. Calculate roots: x1 = 1.2646054962258804, x2 = -0.8896054962258804.
Final ... |
Find the roots of the quadratic equation: 6x^2 + 0x + -7 = 0 | 1. Identify coefficients: a=6, b=0, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -168 = 168.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √168) / 12.
5. Calculate roots: x1 = 1.0801234497346435, x2 = -1.0801234497346435.
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) |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 7^2 = c^2
2. Calculate the squares: 400 + 49 = c^2
3. Sum the squares: 449 = c^2
4. Take the square root of both sides: c = √449
Final Ans... |
Add the fractions 9/9 and 8/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 7, which is 63.
2. Convert 9/9 by multiplying top and bottom by 7: 63/63.
3. Convert 8/7 by multiplying top and bottom by 9: 72/63.
4. Add the numerators: 63 + 72 = 135.
5. The resulting fraction is 135/63.
6. Simplify by d... |
Find the roots of the quadratic equation: 9x^2 + 8x + -1 = 0 | 1. Identify coefficients: a=9, b=8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -36 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √100) / 18.
5. Calculate roots: x1 = 0.1111111111111111, x2 = -1.0.
Final Answer: x = 0.1... |
Multiply the following 2x2 matrices:
A =
[-5, 3]
[-1, -5]
B =
[-2, -3]
[5, 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*-2) + (3*5) = 25
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (3*4) = 27
C21 = (A21 * B11) + (A22 * B21) = (-1*-2) + (-5*5) = -23
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (-5*4) = -17
R... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 6^2 = c^2
2. Calculate the squares: 49 + 36 = c^2
3. Sum the squares: 85 = c^2
4. Take the square root of both sides: c = √85
Final Answer:... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 5^2 = c^2
2. Calculate the squares: 9 + 25 = c^2
3. Sum the squares: 34 = c^2
4. Take the square root of both sides: c = √34
Final Answer: ... |
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) |
Multiply the following 2x2 matrices:
A =
[3, -2]
[-1, -3]
B =
[-1, -4]
[-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) = (3*-1) + (-2*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (-2*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-1*-1) + (-3*-1) = 4
C22 = (A21 * B12) + (A22 * B22) = (-1*-4) + (-3*-2) = 10
... |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 13^2 = c^2
2. Calculate the squares: 100 + 169 = c^2
3. Sum the squares: 269 = c^2
4. Take the square root of both sides: c = √269
Final A... |
Add the fractions 4/12 and 15/4. 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 4/12 by multiplying top and bottom by 1: 4/12.
3. Convert 15/4 by multiplying top and bottom by 3: 45/12.
4. Add the numerators: 4 + 45 = 49.
5. The resulting fraction is 49/12.
Final Answer: 49/... |
Add the fractions 4/14 and 7/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 7, which is 14.
2. Convert 4/14 by multiplying top and bottom by 1: 4/14.
3. Convert 7/7 by multiplying top and bottom by 2: 14/14.
4. Add the numerators: 4 + 14 = 18.
5. The resulting fraction is 18/14.
6. Simplify by div... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 8, 14, 1, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 8 + 14 + 1 + 11) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.2, -1.8, 4.2, -8.8, 1.2]
Squared Differences: [27.04, 3.24, 17.64, 77.44, 1.44]
... |
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) |
Add the fractions 10/4 and 11/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 15, which is 60.
2. Convert 10/4 by multiplying top and bottom by 15: 150/60.
3. Convert 11/15 by multiplying top and bottom by 4: 44/60.
4. Add the numerators: 150 + 44 = 194.
5. The resulting fraction is 194/60.
6. Simpli... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-2, 4]
B =
[2, -5]
[-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) = (-1*2) + (3*-4) = -14
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (3*-1) = 2
C21 = (A21 * B11) + (A22 * B21) = (-2*2) + (4*-4) = -20
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (4*-1) = 6
Res... |
Add the fractions 14/7 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 13, which is 91.
2. Convert 14/7 by multiplying top and bottom by 13: 182/91.
3. Convert 6/13 by multiplying top and bottom by 7: 42/91.
4. Add the numerators: 182 + 42 = 224.
5. The resulting fraction is 224/91.
6. Simplif... |
Compute the quotient and remainder of 3524 divided by 48. Show the long division steps. | To divide 3524 by 48:
1. Determine how many times 48 goes into the leading digits of 3524.
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: 3524 / 48 = 73 with a remainder of 20.
Proof: (73 * 48) + 20 ... |
Compute the quotient and remainder of 44844 divided by 55. Show the long division steps. | To divide 44844 by 55:
1. Determine how many times 55 goes into the leading digits of 44844.
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: 44844 / 55 = 815 with a remainder of 19.
Proof: (815 * 55) ... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^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: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 10, 9, 3, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 10 + 9 + 3 + 18) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, 0.6, -0.4, -6.4, 8.6]
Squared Differences: [5.76, 0.36, 0.16, 40.96, 73.96]
... |
Add the fractions 15/10 and 11/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 7, which is 70.
2. Convert 15/10 by multiplying top and bottom by 7: 105/70.
3. Convert 11/7 by multiplying top and bottom by 10: 110/70.
4. Add the numerators: 105 + 110 = 215.
5. The resulting fraction is 215/70.
6. Simp... |
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 |
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) |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 19^2 = c^2
2. Calculate the squares: 324 + 361 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
Final A... |
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 |
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 |
Find the roots of the quadratic equation: 4x^2 + -9x + -3 = 0 | 1. Identify coefficients: a=4, b=-9, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -48 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √129) / 8.
5. Calculate roots: x1 = 2.5447270864500684, x2 = -0.2947270864500684.
Final A... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Multiply the following 2x2 matrices:
A =
[-4, 5]
[5, -3]
B =
[-2, 3]
[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) = (-4*-2) + (5*0) = 8
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (5*2) = -2
C21 = (A21 * B11) + (A22 * B21) = (5*-2) + (-3*0) = -10
C22 = (A21 * B12) + (A22 * B22) = (5*3) + (-3*2) = 9
Result M... |
Add the fractions 2/12 and 14/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 11, which is 132.
2. Convert 2/12 by multiplying top and bottom by 11: 22/132.
3. Convert 14/11 by multiplying top and bottom by 12: 168/132.
4. Add the numerators: 22 + 168 = 190.
5. The resulting fraction is 190/132.
6. ... |
Add the fractions 1/3 and 11/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 1/3 by multiplying top and bottom by 5: 5/15.
3. Convert 11/5 by multiplying top and bottom by 3: 33/15.
4. Add the numerators: 5 + 33 = 38.
5. The resulting fraction is 38/15.
Final Answer: 38/15 |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 9, 19, 14, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 9 + 19 + 14 + 9) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.2, -2.2, 7.8, 2.8, -2.2]
Squared Differences: [38.44, 4.84, 60.84, 7.84, 4.84]
... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Add the fractions 7/2 and 9/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 5, which is 10.
2. Convert 7/2 by multiplying top and bottom by 5: 35/10.
3. Convert 9/5 by multiplying top and bottom by 2: 18/10.
4. Add the numerators: 35 + 18 = 53.
5. The resulting fraction is 53/10.
Final Answer: 53/1... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 7, 4, 20, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 7 + 4 + 20 + 8) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, -2.4, -5.4, 10.6, -1.4]
Squared Differences: [1.96, 5.76, 29.16, 112.36, 1.96]
... |
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