Example 9. Find all values of $\sqrt[4]{1-i}$.

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Solution. Convert the complex number $1-i$ to trigonometric form

$$
1-i=\sqrt{2}\left[\cos \left(-\frac{\pi}{4}\right)+i \sin \left(-\frac{\pi}{4}\right)\right]
$$

Therefore,

$$
\sqrt[4]{1-i}=\sqrt[8]{2}\left(\cos \frac{-\frac{\pi}{4}+2 k \pi}{4}+i \sin \frac{-\frac{\pi}{4}+2 k \pi}{4}\right)
$$

By setting $k=0,1,2,3$, we find

$$
\begin{aligned}
& (k=0) \quad \sqrt[4]{1-i}=\sqrt[8]{2}\left(\cos \frac{\pi}{16}-i \sin \frac{\pi}{16}\right) \\
& (k=1) \sqrt[4]{1-i}=\sqrt[8]{2}\left(\cos \frac{7}{16} \pi+i \sin \frac{7}{16} \pi\right) \\
& (k=2) \sqrt[4]{1-i}=\sqrt[8]{2}\left(\cos \frac{15}{16} \pi+i \sin \frac{15}{16} \pi\right) \\
& (k=3) \quad \sqrt[4]{1-i}=\sqrt[8]{2}\left(\cos \frac{23}{16} \pi+i \sin \frac{23}{16} \pi\right)
\end{aligned}
$$

## Problems for Independent Solution

Find all values of the root in the following problems:
16. a) $\sqrt[4]{-1}$;
b) $\sqrt{i}$;
c) $\sqrt[3]{i}$;
d) $\sqrt[4]{-i}$
17. a) $\sqrt[4]{1}$;
b) $\sqrt[3]{-1+i}$
c) $\sqrt{2-i 2 \sqrt{3}}$.
18. $\sqrt[5]{\sqrt{2}\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)}$.