How do you binomially expand this?

[coverband]

(x^2+y^2)^0.5

 

[HallsofIvy]

You can look at the "generalized binomial theorem" here:
http://en.wikipedia.org/wiki/Binomial_series

Basically, it says that
$$(x+ y)^\alpha= \sum \left(\begin{array}{c}\alpha \\ i\end{array}\right)x^i y^{\alpha- i}$$
where
$$\left(\begin{array}{c}\alpha \\ i\end{array}\right)= \frac{\alpha(\alpha+ 1)(\alpha- 2)\cdot\cdot\cdot(\alpha- i-1)}{i!}$$
is the "generalized binomial coefficient". If $\alpha$ is a positive integer, the "generalized binomial coefficient" is the usual binomial coefficient and is eventually 0 so the sum is finite. If $\alpha$ is not a positive integer (and for your problem, it is 1/2) the sum is an infinite series.

With $x^2$ and $y^2$ instead of x and y, it just becomes
$$(x^2+ y^2)^\alpha= \sum \left(\begin{array}{c}\alpha \\ i\end{array}\right)x^{2i} y^{2(\alpha- i)}$$