Question

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

Answer 1

Well if you're referring to rationalizing $\bf \cfrac{2}{\sqrt{2}}$, 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  $\bf \cfrac{2}{2}=1\qquad \cfrac{3}{3}=1\qquad \cfrac{1,000,000}{1,000,000}=1 \\\\\\ \cfrac{ducks}{ducks}=1\qquad \cfrac{spaghetti}{spaghetti}=1\qquad \cfrac{cheese}{cheese}=1 \\\\\\ \cfrac{\textit{the quick fox jumped}}{\textit{the quick fox jumped}}=1\qquad \cfrac{whatever}{whatever}=1$

so.. when you're doing $\bf \cfrac{2}{\sqrt{2}}\cdot \cfrac{\sqrt{2}}{\sqrt{2}}\iff \cfrac{2}{\sqrt{2}}\cdot1$

and the value multiplicand doesn't change in any way

now, try this in your calculator  $\bf \cfrac{2}{\sqrt{2}}\cdot \cfrac{\sqrt{2}}{\sqrt{2}}\implies \cfrac{2\sqrt{2}}{\sqrt{2^2}}\implies \cfrac{2\sqrt{2}}{2}\implies \sqrt{2}\\\\ -----------------------------\\\\ \textit{check how much is }\cfrac{2}{\sqrt{2}}\textit{ in your calculator} \\\\\\ \textit{then check how much is }\sqrt{2}\textit{ in it}$