Answer
$\frac{1}{\sqrt{x+1}+\sqrt{x}}$
Work Step by Step
We simplify the fraction by multiplying through by $\sqrt{x+1}+\sqrt{x}$ and using the fact that $(a-b)(a+b)=a^2-b^2$:
$\displaystyle \sqrt{x+1}-\sqrt{x}=\frac{\sqrt{x+1}-\sqrt{x}}{1}*\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}=\frac{x+1-x}{\sqrt{x+1}+\sqrt{x}}=\frac{1}{\sqrt{x+1}+\sqrt{x}}$
