Question

CMR :

\(\left(\dfrac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right).\left(\dfrac{1-\sqrt{a}}{1-a}\right)^2=1\)

Answer 1

\(VP=\left(\dfrac{1-\left(\sqrt{a}\right)^3}{1-\sqrt{a}}+\sqrt{a}\right)\left(\dfrac{1-\sqrt{a}}{1-\left(\sqrt{a}\right)^2}\right)^2\)

\(=\left(\dfrac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right)\left(\dfrac{1-\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\right)^2\)

\(=\left(1+2\sqrt{a}+a\right)\left(\dfrac{1}{1+\sqrt{a}}\right)^2\)

\(=\left(1+\sqrt{a}\right)^2.\left(\dfrac{1}{\sqrt{a}+1}\right)^2=\left(\dfrac{1+\sqrt{a}}{1+\sqrt{a}}\right)^2=1\) (đpcm)