Question

a>0, b>0 ,a≠b

chứng minh ( \(\sqrt{\dfrac{a}{b}}\)-\(\sqrt{\dfrac{b}{a}}\)) :(a-b) = \(\dfrac{1}{\sqrt{ab}}\)

mn giúp e với ạ

Answer 1

\(VT=\left(\sqrt{\dfrac{a}{b}}-\sqrt{\dfrac{b}{a}}\right):\left(a-b\right)\\ =\left(\dfrac{\sqrt{a}}{\sqrt{b}}-\dfrac{\sqrt{b}}{\sqrt{a}}\right).\dfrac{1}{a-b}\\ =\dfrac{\sqrt{a}.\sqrt{a}-\sqrt{b}.\sqrt{b}}{\sqrt{ab}}.\dfrac{1}{a-b}\\ =\dfrac{\sqrt{a^2}-\sqrt{b^2}}{\sqrt{ab}}.\dfrac{1}{a-b}\\ =\dfrac{a-b}{\sqrt{ab}}.\dfrac{1}{a-b}\\ =\dfrac{1}{\sqrt{ab}}=VP\left(dpcm\right)\)

Answer 2

\(VT=\dfrac{a-b}{\sqrt{ab}}\cdot\dfrac{1}{a-b}=\dfrac{1}{\sqrt{ab}}=VP\)