### Home > PC3 > Chapter 11 > Lesson 11.2.5 > Problem11-152

11-152.

What are the four fourth roots of $−16$?

Let $z = −16$. Write $z$ in polar form.

Given the complex number $z = r ( \cos \theta ) + i \sin ( \theta ) )$, the $\boldsymbol n^{\bf th}$ roots of $z$ are given by:
${ \sqrt [ n ] { r } ( \operatorname { cos } ( \frac { \theta + 2 k \pi } { n } ) + i \operatorname { sin } ( \frac { \theta + 2 k \pi } { n } ) ) \quad \text { for } k = 0,1,2 }$

$z = 16(\cos(π) + i\sin(π))$
$r = 16$
$n = 4$
$θ = π$