Find $3^{2-i}$ in the form x+yi

Question

Find $3^{2-i}$ in the form x+yi

How do I do this question?

$e^{\ln3}$$^{^{2-i}}$ Is that right so far?

Answer 1

Yes, and then $\large e^{\ln(3^{2-i})}=e^{(2-i)\ln3}=e^{2\ln 3}(\cos(\ln 3)-i\sin(ln 3))$

So you can have it in the form $a+bi$.

Answer 2

Yes you are on the right track. We have,

$$3^{2-i} = e^{(2-i)\ln{3}} = e^{2\ln{3}} e^{-i\ln{3}}=e^{2\ln{3}}\left(\cos{(-\ln{3}) + i\sin{(-\ln{3}}} \right) = \ldots$$