Question

Chứng minh đẳng thức:

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

Answer 1

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

\(\Leftrightarrow\dfrac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}=\dfrac{\left(\sqrt{a^3}+\sqrt{b^3}\right)}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}=\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}=a-\sqrt{ab}+b-\sqrt{ab}=a-2\sqrt{ab}+b=\left(\sqrt{a}-\sqrt{b}\right)^2\)

\(\Rightarrow\text{đ}pcm\)