Answer:
1
Step-by-step explanation:
You want to simplify the given complex fraction involving fractional powers.
Simplify
We can define a = x^(1/2) and b = y^(1/2) to remove fractional powers from the expression. The rest of it involves factoring the sum of cubes and the difference of squares. Common factors are cancelled.
[tex]\left(\dfrac{a^3+b^3}{a^2-b^2}-\dfrac{a^2}{a+b}-\dfrac{b^2}{a-b}\right)\div\dfrac{ab}{a+b}\\\\=\left(\dfrac{a^3+b^3}{a^2-b^2}-\dfrac{a^2}{a+b}-\dfrac{b^2}{a-b}\right)\times\dfrac{a+b}{ab}\\\\=\left(\dfrac{(a^3+b^3)-a^2(a-b)-b^2(a+b)}{a^2-b^2}\right)\times\dfrac{a+b}{ab}\\\\=\dfrac{(a^3+b^3-a^3+a^2b-ab^2-b^3)(a+b)}{ab(a-b)(a+b)}=\dfrac{a^2b-ab^2}{ab(a-b)}=\dfrac{a^2b-ab^2}{a^2b-ab^2}\\\\=\boxed{1}[/tex]
Note that there are no instances of 'a' or 'b' left in the expression, so we do not need to back-substitute for 'a' and/or 'b' in the simplified form.
