Why is it mathematically legal to multiply 2 √ 2 by √ 2 √ 2 in order to rationalize the denominator?

Well if you're referring to rationalizing , which simply means, getting rid of the pesky radical at the bottom

well, it boils down to, hmm say... a quantity or even a polynomial, multiplied times 1, is itself, 2*1=2, 3*1 = 3, ducks*1 = ducks, spaghetti * 1 = spaghetti

or whatever * 1 = whatever
and the value of the multiplicand, doesn't change in anyway, is the same thing before and after the multiplication by 1

now....1 can also be a fraction  

so.. when you're doing 

and the value multiplicand doesn't change in any way

now, try this in your calculator