# Simplifying an expression involving square roots and finding a result

$$\left(\frac{1}{\sqrt a + \sqrt {a+1}} + \frac{1}{\sqrt a - \sqrt {a-1}}\right):\left(1+\frac{\sqrt{a+1}}{\sqrt{a-1}} \right)$$

I have tried simplifying the denominators, with no success. I was thinking about doing it the hard way and just multiplying both terms in the first parenthesis to get a common denominator, but I was wondering if there is an easier way to solve this.

• Does the "$:$" denote division? – angryavian Jan 13 '17 at 2:48
• yes, the : symbol is division. – Rick Sanchez Jan 13 '17 at 2:49

$\displaystyle \frac{1}{\sqrt{a} + \sqrt{a+1}} + \frac{1}{\sqrt{a}-\sqrt{a-1}} = \sqrt{a+1}-\sqrt{a} + \sqrt{a} +\sqrt{a-1}=\sqrt{a+1} + \sqrt{a-1}$, so $\displaystyle (\sqrt{a+1} + \sqrt{a-1})\cdot\frac{\sqrt{a-1}}{\sqrt{a+1} + \sqrt{a-1}} = \sqrt{a-1}$