Evaluate (-256)^(1/4)

Problem

(−256)(1/4)

Solution

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

2. Apply the property of roots for negative bases, noting that the fourth root of a negative number is not a real number.

3. Rewrite the expression using the imaginary unit i where i=√(,−1)

4. Factor the expression into the real fourth root and the fourth root of −1

(−256)(1/4)=(256⋅−1)(1/4)

5. Calculate the fourth root of 256 which is 4 because 4=256

(256)(1/4)=4

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

−1=e(i*π)

(−1)(1/4)=(e(i*π))(1/4)=e(i*π/4)

7. Convert the result to rectangular form.

e(i*π/4)=cos(π/4)+i*sin(π/4)

e(i*π/4)=√(,2)/2+√(,2)/2*i

8. Multiply the real root by the complex component to find the principal value.

4⋅(√(,2)/2+√(,2)/2*i)=2√(,2)+2√(,2)*i

Final Answer

(−256)(1/4)=2√(,2)+2√(,2)*i

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