Answer:
Option C.
Step-by-step explanation:
The given expression is
[tex]\left(\sqrt{2}cis \dfrac{9\pi}{20}\right)^5[/tex]
It can be rewritten as
[tex]\left(\sqrt{2}\cos \dfrac{9\pi}{20}+i\sin\dfrac{9\pi}{20}\right)^5[/tex]
According to De moivre's Theorem:
[tex][r(\cos\theta+i\sin \theta)]^n=r^n(\cos n\theta +i\sin n\theta)[/tex]
Using De moivre's Theorem, we get
[tex](\sqrt{2})^5\left(\cos \dfrac{9\pi\times 5}{20}+i\sin\dfrac{9\pi\times 5}{20}\right)[/tex]
[tex]=4\sqrt{2}\left(\cos \dfrac{9\pi}{4}+i\sin\dfrac{9\pi}{4}\right)[/tex]
[tex]=4\sqrt{2}\left(\cos (2\pi +\dfrac{\pi}{4})+i\sin (2\pi+\dfrac{\pi}{4})\right)[/tex]
[tex]=4\sqrt{2}\left(\cos \dfrac{\pi}{4}+i\sin \dfrac{\pi}{4}\right)[/tex]
[tex]=4\sqrt{2}\left(\dfrac{1}{\sqrt{2}}+i\dfrac{1}{\sqrt{2}}\right)[/tex]
[tex]=4+4i[/tex]
Therefore, the correct option is C.
