Evaluate (-16)^(1/4)

Problem

(−16)(1/4)

Solution

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

2. Apply the property of roots for negative numbers using the imaginary unit i where i2=−1

3. Rewrite the expression using the complex polar form or by recognizing that the fourth root of a negative number results in complex values.

4. Use the formula for the nth roots of a complex number z=r*(cos(θ)+i*sin(θ)) where r=16 and θ=π

5. Calculate the principal root using w=r(1/n)*(cos(θ/n)+i*sin(θ/n))

6. Substitute the values r=16 n=4 and θ=π to get 16(1/4)*(cos(π/4)+i*sin(π/4))

7. Simplify the constants and trigonometric values where 16(1/4)=2 cos(π/4)=√(,2)/2 and sin(π/4)=√(,2)/2

8. Distribute the 2 to find the principal value.

Final Answer

(−16)(1/4)=√(,2)+i√(,2)

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