# Write this complex number on the form a + bi

• Oct 13th 2012, 01:41 PM
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.

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

I've tried writing it as

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

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

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

• Oct 13th 2012, 01:56 PM
HallsofIvy
Re: Write this complex number on the form a + bi
You suspect have misread the answer you are supposed to get. It should be $2\sqrt{5}/5+ i\sqrt{5}/5$ or $2\frac{\sqrt{5}}{5}+ i\frac{\sqrt{5}}{5}$ either of which is equal to $\frac{2}{\sqrt{5}}+ \frac{i}{\sqrt{5}}$.

You get that by, as you say, multiplying by the conjugate. Then "rationalize the denominator".
• Oct 13th 2012, 02:12 PM
Plato
Re: Write this complex number on the form a + bi
Quote:

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