# Find the fourth roots of -17i?

May 23, 2016

Fourth roots of $- 17 i$ are $\sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$

#### Explanation:

Let us wrote $- 17 i$ in polar form and as its modulus is $17$ and number is $117 \times \left(- i\right)$, we can write it as

$- 17 i = 17 {e}^{\frac{3 \pi}{2} i}$

Let $x = \sqrt[4]{- 17 i}$, then

${x}^{4} = - 17 i = 17 {e}^{\frac{3 \pi}{2} i}$

Hence $x = \sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$

Other roots can be obtained by adding $\frac{\pi}{2}$ to $\frac{3 \pi}{8}$ in succession (as they are one-fourth of $2 \pi$).

i.e. $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$

To find them in the form of $a + b i$, you can use the following

$r {e}^{i \theta} = r \cos \theta + i \sin \theta$