This is so we can apply the difference of squares rule in step 3. The difference of squares rule is (x-y)(x+y) = x^2-y^2.In step 4, the square roots cancel out with the squaring operation. The two operations are inverses of one another, which is why they cancel.
and you want to rationalize the denominator, then you should multiply top and bottom by 
. The same applies in reverse as well.
and 
What is the following quotient?
...
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