# Simplify the complex fraction: [(√x-√y)/(√x+√y)-(√x+√y)/(√x-√y)] / [(√x-√y)/(√x+√y)+(√x+√y)/(√x-√y)]Show complete solution.

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You must find out the common denominator of fractions from numerator. This common denominator is (√x+√y)(√x-√y). This special product is the difference of squares x-y.

You must multiply the fractions from numerator by the common denominator.

[(√x-√y)^2 - (√x+√y)^2]/(x-y)

You will expand the squares with formula:

(a+b)^2=a^2+2ab+b^2

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

(√x-√y)^2=x-2√xy+y

(√x+√y)^2=x+2√xy+y

(√x-√y)^2 - (√x+√y)^2=x-2√xy+y-x-2√xy-y=-4√xy

You will repeat the steps to calculate the addition from denominator.

[(√x-√y)^2 + (√x+√y)^2]=x-2√xy+y+x+2√xy+y=2x+2y=2(x+y)

You will write the resulting fraction:

[-4√xy/(x-y)]/[2(x+y)/(x-y)]

Multiply the fraction from numerator by the reversed fraction from denominator.

[-4√xy/(x-y)]*[(x-y)/2(x+y)]

Simplify by x-y => [-4√xy/2(x+y)] = -2√xy/(x+y)

**Answer: The result of fraction simplification is -2√xy/(x+y).**