I'm dealing with this problem where I'm to write the complex number below on the form a + bi.

$\displaystyle \frac {\sqrt{2+i}}{\sqrt{2-i}}$

I've tried writing it as

$\displaystyle \sqrt {\frac {2+i}{2-i}}$

Then I tried multiplying by the conjugate, writing it using polar coordinates, considering the squre root as an exponent of 0.5 etc. However, I just kept reaching a dead end. The expression is supposedly equal to

$\displaystyle \fr {2\sqrt{5}}{5} + \fr {i\sqrt{5}}{5}$

How do I go about?