Question

RG: \(\dfrac{a\sqrt{a}-b\sqrt{b}+a\sqrt{b}-b\sqrt{a}}{a+b+2\sqrt{ab}}\)

Answer 1

\(\dfrac{a\sqrt{a}-b\sqrt{b}+a\sqrt{b}-b\sqrt{a}}{a+b+2\sqrt{ab}}\)

\(=\dfrac{\left(a\sqrt{a}+a\sqrt{b}\right)-\left(b\sqrt{b}+b\sqrt{a}\right)}{\left(\sqrt{a}+\sqrt{b}\right)^2}\)

\(=\dfrac{a\left(\sqrt{a}+\sqrt{b}\right)-b\left(\sqrt{b}+\sqrt{a}\right)}{\left(\sqrt{a}+\sqrt{b}\right)^2}\)

\(=\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(a-b\right)}{\left(\sqrt{a}+\sqrt{b}\right)^2}\)

\(=\dfrac{a-b}{\sqrt{a}+\sqrt{b}}\)

\(=\dfrac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}+\sqrt{b}}\)

\(=\sqrt{a}-\sqrt{b}\)

\(Ayumu\)