# Expand the following expression : $\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+y^2)} (x+\sqrt{(x^2+y^2)}))}$

I have the following expression :

$$\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+y^2)} (x+\sqrt{(x^2+y^2)}))}$$

I don't understand why :

$$\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+y^2)} (x+\sqrt{(x^2+y^2)}))}=\frac{1}{\sqrt{x^2+y^2}}$$

I tried I got this :

$$\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{1}{x+\sqrt{x^2+y^2}}*\frac{x}{\sqrt{x^2+y^2}}$$

Yet, I don't get the required result which is : $$\frac{1}{\sqrt{x^2+y^2}}$$

I don't understand what I missing in the expanding process. Any ideas?

Thank you.

• It's easy. you are almost done. Try a little more.
– user249332
Sep 8, 2015 at 16:44

Let me try. $$\frac{1}{x+\sqrt{x^2+y^2}} + \frac{1}{x+\sqrt{x^2+y^2}}\frac{x}{\sqrt{x^2+y^2}} = \frac{1}{x+\sqrt{x^2+y^2}}\left( 1 + \frac{x}{\sqrt{x^2+y^2}}\right) = \frac{1}{x+\sqrt{x^2+y^2}}\frac{x+\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}} = \frac{1}{\sqrt{x^2+y^2}}.$$