### Can you help me figure out this problem?

It's driving me crazy. I was working with my Geometry students today, and I got stuck on a problem. We were on the last step. I knew what the answer was because I had the teacher's guide, but I couldn't for the life of me figure out how it went from the previous step to the final answer. Even after looking up simplifying radicals in a couple of different textbooks, and trying to find the info with a Google search, I'm still stumped. I thought perhaps some of the smart people on my flist could explain it to me. :D

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

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