Write this complex number on the form a + bi

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?

Re: Write this complex number on the form a + bi

You suspect have misread the answer you are supposed to get. It should be $\displaystyle 2\sqrt{5}/5+ i\sqrt{5}/5$ or $\displaystyle 2\frac{\sqrt{5}}{5}+ i\frac{\sqrt{5}}{5}$ either of which is equal to $\displaystyle \frac{2}{\sqrt{5}}+ \frac{i}{\sqrt{5}}$.

You get that by, as you say, multiplying by the conjugate. Then "rationalize the denominator".

Re: Write this complex number on the form a + bi

Quote:

Originally Posted by

**MathCrusader** 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

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

How do I go about?

$\displaystyle \frac {\sqrt{2+i}}{\sqrt{2-i}}\frac {\sqrt{2-i}}{\sqrt{2-i}}=\frac {\sqrt{5}}{2-i}=\frac {2\sqrt{5}+\sqrt{5}i}{5}$