Here is the problem:

Find the geometric mean of 4√3 and 5√2.

According to my teacher's guide, you set up the proportion, cross multiply, and isolate the variable. At that point you get:

x=√(4√3 * 5√2) <--All that is under the first radical.

Now, I can think of one way to deal with "square root of square root of…": change to fractional exponents. But that only makes the problem messier, as far as I can tell. I can also use the calculator to approximate square roots, which gives the answer of around 6.98. But my answer key says that the final answer is the whole number 7, so there must be a better way to simplify it. Do you know what it is?

Joie