Evaluate the sum \[\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dotsb + \frac{1}{\sqrt{15} + \sqrt{16}}.\]

(You might want to use a calculator to get an estimate of the answer, but in order to get the exact answer you will have to do it by hand!)

The denominators are quite scary. Maybe we can make them nicer somehow.

Things like \(\sqrt{5}+\sqrt{6}\) make me think of the difference of two squares….