Question

Cho biểu thức \(A=\left(\frac{1}{a-\sqrt{a}}+\frac{1}{\sqrt{a}-1}\right):\frac{\sqrt{a+1}}{a-2\sqrt{a}+1}\) ( với \(a>0\) và \(a\ne1\) ).

a ) Rút gọn biểu thức A

b ) Tìm a để \(A=\frac{1}{2}\)

Answer 1

a) \(A=\left(\frac{1}{a-\sqrt{a}}+\frac{1}{\sqrt{a}-1}\right):\frac{\sqrt{a}+1}{a-2\sqrt{a}+1}\left(a>0;a\ne1\right)\)

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

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

\(=\frac{\sqrt{a}-1}{\sqrt{a}}\)

b) Để \(A=\frac{1}{2}\)

\(\Leftrightarrow\frac{\sqrt{a}-1}{\sqrt{a}}=\frac{1}{2}\)

\(\Leftrightarrow2\sqrt{a}-2=\sqrt{a}\)

\(\Leftrightarrow\sqrt{a}=2\Leftrightarrow a=4\left(tm\right)\)