simplifying a fraction

Question

I got stuck on a problem that I found the answer to but it is bugging me because I don't understand how they simplified a piece of the problem. It was 1/(sqrt2/2)^3. I see it simplifies to 2sqrt2 but I dont know how they got the answer.

Answer 1

This is a fraction problem that can be solved in two steps:

1) Simplify the denominator: 1/(sqrt2/2)³ = 1/(sqrt2)³/8 = 8/(sqrt2)³ (fractions flipped - double division)

2) Now, since we know that

(sqrt 2)³ = sqrt 2 . sqrt 2, sqrt 2 (3 times in multiplication)

& (sqrt 2) . (sqrt 2) = 2

thus,

8/ (sqrt2)^3 = 8/2.sqrt2 = 4/sqrt2 = 2 sqrt 2