http://i53.tinypic.com/vgr8yp.jpg
^see pic
Thank you so much!!
http://i53.tinypic.com/vgr8yp.jpg
^see pic
Thank you so much!!
They've multiplied by a cleverly disguised $\displaystyle 1$.
In this case, $\displaystyle \displaystyle\frac{\frac{1}{x}}{\frac{1}{x}}$.
So $\displaystyle \displaystyle{\frac{x}{\sqrt{x^2 + x} + x} = \frac{x}{\sqrt{x^2+x}+x}\cdot \frac{\frac{1}{x}}{\frac{1}{x}}}$
$\displaystyle \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}+x}{x}}}$
$\displaystyle \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}}{x} + 1}}$
$\displaystyle \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}}{\sqrt{x^2}} + 1}}$
$\displaystyle \displaystyle{= \frac{1}{\sqrt{\frac{x^2 + x}{x^2}} + 1}}$
$\displaystyle \displaystyle{= \frac{1}{\sqrt{1 + \frac{1}{x}} + 1}}$.