How do you rationalize the denominator and simplify $(sqrtx + 2sqrty)/(sqrtx - 2sqrty)$ ?

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Answer 1 · 2016-04-03T18:31:49

$(x+4sqrt(xy)+4y)/(x-4y)$

Multiply the fraction by the conjugate of its denominator.

$=(sqrtx+2sqrty)/(sqrtx-2sqrty)((sqrtx+2sqrty)/(sqrtx+2sqrty))$

In the denominator, we have what will become a difference of squares, which take the form:

$(a+b)(a-b)=a^2-b^2$

Here, we have $a=sqrtx$ , so $a^2=x$ , and $b=2sqrty$ , which implies that $b^2=4y$ .

Thus, the fraction simplifies to be

$=(sqrtx+2sqrty)^2/(x-4y)$

This is a simplified answer. However, another perfectly acceptable way of presenting this would be to distribute the squared binomial as follows:

$=((sqrtx+2sqrty)(sqrtx+2sqrty))/(x-4y)=(x+4sqrt(xy)+4y)/(x-4y)$

How do you rationalize the denominator and simplify (sqrtx + 2sqrty)/(sqrtx - 2sqrty)(sqrtx + 2sqrty)/(sqrtx - 2sqrty)?