Thank you for walking me through this problem step-by-step.
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Hey mlaboda:
Thanks for posting this - this is one of my favorite problems (I had a student bring it into a class once).
I may let the community work through this one instead of just popping in with the right answer, but a few guiding principles to get started (I'll check in later to make sure it gets answered):
1) Deal with first thing's first. Because this term is squared, you need to get rid of the parentheses by squaring it and listing out the individual terms.
2) To eliminate square roots, you have to multiply, so you need to look for opportunities to multiply a root by another to "square out the root".
3) When multiplying roots, you can combine them under one radical sign (e.g. [sqrt (x)] * [sqrt (y)] = sqrt (xy) ), which helps to streamline the number of roots you're working with.
4) When you see a set up like this one with an x+y and an x-y (this one has 7 +sqrt 48 and 7-sqrt 48) there's a 90%+ chance that you'll be able to use that Difference of Squares algebra rule: (x+y)(x-y) = x^2 - y^2, so that's something you should look for.
With those principles, you should be able to work through this one. I'll let the community work it through (learning by doing is much more powerful than learning by being told how to do) and check in later today to help out as needed.
