How does $\sqrt{n+1}-\sqrt{n} = \frac{(n+1) - n}{\sqrt{n+1}+\sqrt{n}}$? How does $\sqrt{n+1}-\sqrt{n} = \frac{(n+1) - n}{\sqrt{n+1}+\sqrt{n}}$  ?
What are the exact steps to get to the right side from the left?

It's the hint with almost solution and quite good technic for such problems.
$$ \bigg( \sqrt{n+1}-\sqrt{n} \bigg) \cdot 1 = \bigg( \sqrt{n+1}-\sqrt{n} \bigg) \cdot \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}$$
Useful formula for this task:
$$ \bigg( \sqrt{a} - \sqrt{b} \bigg) \cdot \bigg( \sqrt{a} + \sqrt{b} \bigg) = a - b $$
