Simplifying Square Roots within a Square Root

I recently wrote a Math Team problem for my school to be turned in to the league as our contribution to the test bank for future Math competition. Every year I am asked by the Math League coach to write a Math Team problem. This year I decided to write a problem that may not seem as obvious at first thought, but after a second glance is a nice method to making a problem easier. The problem below depicts the technique used to solving the equation I wrote for the Math competition. The problem below was not the problem I turned in, but it has similar techniques to solve square root equation problems. Here is a simple trick of the trade when simplifying square roots within a square root:

Creating a perfect square to remove the larger square root on the surface may not be so obvious at first glance, but you can see that this techniques presents the fastest and easiest approach to removing the larger square root. With this template, anyone can create a large amount of problems that could be used as a bonus question on a test or quiz. The goal is to pick consecutive numbers such as 26 and 27 to be placed within the square root as shown in the example above. Replace the above problem with 107 and 108. You will see that the answer will be nicely determined based on the technique. I hope this helps a little with understanding simplifying certain types of square root problem.

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