what is as a complex number?

\(\text{do you mean}\\ \sqrt{2^{i7\pi/4}}\)

.

\(y=(\sqrt2)^{(\frac{7\pi i}{4})}\\ y=2^{(\frac{7\pi }{8}i)}\\ log y=log 2^{(\frac{7\pi }{8}i)}\\ ln y=(\frac{7\pi }{8}i) ln 2\\ e^{ln y}=e^{(\frac{7ln 2\pi }{8}i) }\\ y=e^{(\frac{7(ln 2)\pi }{8})i }\\ y=cos(\frac{7(ln 2)\pi }{8})+isin(\frac{7(ln 2)\pi }{8})\)

 

 

You had better check that very carefully.