Rút gọn các biểu thức sau
1) $\frac{1}{x+\sqrt[2]{x} }$ + $\frac{2\sqrt[2]{x}}{x-1}$ – $\frac{1}{x- \sqrt[2]{x} }$ ( với x>0; x $\neq$ 1
Rút gọn các biểu thức sau
1) $\frac{1}{x+\sqrt[2]{x} }$ + $\frac{2\sqrt[2]{x}}{x-1}$ – $\frac{1}{x- \sqrt[2]{x} }$ ( với x>0; x $\neq$ 1
Đáp án:
$\begin{array}{l}
Dkxd:x > 0;x \ne 1\\
A = \dfrac{1}{{x + \sqrt x }} + \dfrac{{2\sqrt x }}{{x – 1}} – \dfrac{1}{{x – \sqrt x }}\\
= \dfrac{1}{{\sqrt x \left( {\sqrt x + 1} \right)}} + \dfrac{{2\sqrt x }}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right)}} – \dfrac{1}{{\sqrt x \left( {\sqrt x – 1} \right)}}\\
= \dfrac{{\sqrt x – 1 + 2\sqrt x .\sqrt x – \left( {\sqrt x + 1} \right)}}{{\sqrt x \left( {\sqrt x + 1} \right)\left( {\sqrt x – 1} \right)}}\\
= \dfrac{{x – 1 + 2x – \sqrt x – 1}}{{\sqrt x \left( {\sqrt x + 1} \right)\left( {\sqrt x – 1} \right)}}\\
= \dfrac{{3x – \sqrt x – 2}}{{\sqrt x \left( {\sqrt x + 1} \right)\left( {\sqrt x – 1} \right)}}\\
= \dfrac{{\left( {3\sqrt x – 2} \right)\left( {\sqrt x + 1} \right)}}{{\sqrt x \left( {\sqrt x + 1} \right)\left( {\sqrt x – 1} \right)}}\\
= \dfrac{{3\sqrt x – 2}}{{\sqrt x \left( {\sqrt x – 1} \right)}}\\
= \dfrac{{3\sqrt x – 2}}{{x – \sqrt x }}
\end{array}$