Question

Rút gọn \(P=\left(\frac{a\sqrt{a}-1}{a-\sqrt{a}}-\frac{a\sqrt{a}+1}{a+\sqrt{a}}\right):\frac{a+2}{a-2}\); a > 0 ; a \(\ne\) 1 ; 2

tìm gia trị nguyên của a để P có giá trị nguyên

Answer 1

\(P=\left[\frac{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}-\frac{\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}+1\right)}\right]:\frac{a+2}{a-2}\)

\(=\left(\frac{a+\sqrt{a}+1}{\sqrt{a}}-\frac{a-\sqrt{a}+1}{\sqrt{a}}\right):\frac{a+2}{a-2}\)\(=\frac{a+\sqrt{a}+1-a+\sqrt{a}-1}{\sqrt{a}}:\frac{a+2}{a-2}\)\(=\frac{2\sqrt{a}}{\sqrt{a}}.\frac{a-2}{a+2}=\frac{2a-4}{a+2}\)