Find the indicated root.

To find the eighth root of , we can express in its polar form. In the complex plane, can be written as (since it lies on the negative x-axis). Now, when seeking the eighth roots, we apply De Moivre’s Theorem, which states that the th roots of a complex number can be found using:

For , and , and we are looking for the roots. Thus, the eighth roots can be calculated as:

where .

Calculating for each :

These provide the eight distinct eighth roots of . The final results are: