Question

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

Answer 1

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

Explanation:

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)`