Evaluate (-81)^(1/4)

Problem

(−81)(1/4)

Solution

1. Identify the expression as the fourth root of a negative number.

2. Rewrite the expression using the imaginary unit i where i2=−1

3. Express the negative number as a product of the positive value and −1

(−81)(1/4)=(81⋅−1)(1/4)

4. Apply the power of a product rule to separate the terms.

(81)(1/4)⋅(−1)(1/4)

5. Calculate the fourth root of 81 which is 3 because 3=81

3⋅(−1)(1/4)

6. Determine the principal fourth root of −1 using Euler's formula, e(i*θ)=cos(θ)+i*sin(θ) Since −1=e(i*π) the principal root is e(i*π/4)

e(i*π/4)=cos(π/4)+i*sin(π/4)

7. Substitute the trigonometric values cos(π/4)=√(,2)/2 and sin(π/4)=√(,2)/2

(−1)(1/4)=√(,2)/2+√(,2)/2*i

8. Multiply the result by 3 to find the principal value.

3⋅(√(,2)/2+√(,2)/2*i)=(3√(,2))/2+(3√(,2))/2*i

Final Answer

(−81)(1/4)=(3√(,2))/2+(3√(,2))/2*i

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