To evaluate the expression \((-8)^{1/3}\), we are looking for the cube root of \(-8\).
1. Formulate the Problem:
The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). Here, we need to find the number \(y\) such that \(y^3 = -8\).
2. Understanding Cube Roots of Negative Numbers:
For negative numbers, the cube root can have both a real and an imaginary component.
3. Calculation:
The cube root of \(-8\) can be complex. When dealing with complex numbers, the principal cube root takes into account the magnitude and argument (angle) in the complex plane.
4. Result:
The result of the cube root of \(-8\) is:
[tex]\[
(1.0000000000000002 + 1.7320508075688772j)
\][/tex]
In conclusion, the correct choice for evaluating \((-8)^{1/3}\) is:
[tex]\[
(1.0000000000000002 + 1.7320508075688772j)
\][/tex]
