Why does root( 4 + root7) -root( 4 - root(7)) equal root 2?

Question

This problem was on a sample sheet for a math bee, and I couldn't figure out the solution to this problem.

Answer 1

Hi Ninja, since you're told some roots disappear, pretend the first expression can be written without nested roots. It probably isn't a whole number, but perhaps it can be written as a whole number a plus a multiple of sqrt(7), b sqrt(7).

By expanding the left side and comparing it to the right, you get a^2 + b^2 = 4 and 2ab = 1, and you can use substitution to get possible values for a and b.

Once the nested roots are gone, can you think of a next step?