Question

\(\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}+\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}-1\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}-\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}+1\right)\)

rút gọncho em với

Answer 1

ĐKXĐ \(a\ge0,b\ge0\)

\(\Rightarrow\left(\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{ab}-1\right)+\left(\sqrt{ab}+\sqrt{a}\right)\left(\sqrt{ab}+1\right)-\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}\right):\left(\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{ab}-1\right)-\left(\sqrt{ab}+\sqrt{a}\right)\left(\sqrt{ab}+1\right)+\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\right)\)

=\(\left(\dfrac{a\sqrt{b}-\sqrt{a}+\sqrt{ab}-1+ab+\sqrt{ab}+a\sqrt{b}+\sqrt{a}-ab+1}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}\right):\left(\dfrac{a\sqrt{b}-\sqrt{a}+\sqrt{ab}-1-ab-\sqrt{ab}-a\sqrt{b}-\sqrt{a}+ab-1}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\right)\)

=\(\left(\dfrac{2a\sqrt{b}+2\sqrt{ab}}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\right):\left(\dfrac{-2\sqrt{a}-2}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\right)\) 

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

Answer 2

Ta có: \(\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}+\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}-1\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}-\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}+1\right)\)

\(=\left(\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{ab}-1\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}+\dfrac{\left(\sqrt{ab}+\sqrt{a}\right)\left(\sqrt{ab}+1\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}-1\right):\left(\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{ab}-1\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}-\dfrac{\left(\sqrt{ab}+\sqrt{a}\right)\left(\sqrt{ab}+1\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}+1\right)\)

\(=\left(\dfrac{a\sqrt{b}-\sqrt{a}+\sqrt{ab}-1+ab+\sqrt{ab}+a\sqrt{b}+\sqrt{a}}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}-1\right):\left(\dfrac{a\sqrt{b}-\sqrt{a}+\sqrt{ab}-1-\left(ab+\sqrt{ab}+a\sqrt{b}+\sqrt{a}\right)}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}+1\right)\)

\(=\left(\dfrac{2a\sqrt{b}+2\sqrt{ab}+ab-1}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}-1\right):\left(\dfrac{-2\sqrt{a}-1-ab}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}+1\right)\)

\(=\dfrac{2a\sqrt{b}+2\sqrt{ab}}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}:\dfrac{-2\sqrt{a}-2}{\left(\sqrt{ab}+1\right)\left(\sqrt{ab}-1\right)}\)

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

\(=-\sqrt{ab}\)