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}{√(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}{√(2)}\cdot \cfrac{√(2)}{√(2)}\iff \cfrac{2}{√(2)}\cdot1`

and the value multiplicand doesn't change in any way

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