How to solve $(sqrtx+sqrty)/(x^(3/2)+y^(3/2))$ ?

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Answer 1 · 2016-11-29T09:27:10

$(x+sqrt(xy)+y)/(x^2+xy+y^2)$

Using the polynomial identity

$(a^3+b^3)/(a+b)=a^2-a b+b^2$ and calling $a=sqrt(x)$ and $b=sqrt(y)$ we have

$(sqrtx+sqrty)/(x^(3/2)+y^(3/2)) = 1/(x-sqrt(xy)+y)$ and rationalizing

$(sqrtx+sqrty)/(x^(3/2)+y^(3/2)) =(x+sqrt(xy)+y)/((x+y)^2-xy)=(x+sqrt(xy)+y)/(x^2+xy+y^2)$

How to solve (sqrtx+sqrty)/(x^(3/2)+y^(3/2))(sqrtx+sqrty)/(x^(3/2)+y^(3/2)) ?