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ĐKXĐ:....
\(A=\left(\frac{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}{1-\sqrt{a}}+\sqrt{a}\right)\left(\frac{1-\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\right)^2\)
\(A=\left(a+2\sqrt{a}+1\right)\frac{1}{\left(1+\sqrt{a}\right)^2}=\frac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)^2}=1\)
\(B=\frac{2}{\sqrt{ab}}:\left(\frac{\sqrt{b}-\sqrt{a}}{\sqrt{ab}}\right)^2-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(B=\frac{2}{\sqrt{ab}}.\frac{\sqrt{ab}^2}{\left(\sqrt{a}-\sqrt{b}\right)^2}-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}=\frac{2\sqrt{ab}-a-b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(B=\frac{-\left(\sqrt{a}-\sqrt{b}\right)^2}{\left(\sqrt{a}-\sqrt{b}\right)^2}=-1\)
a. ĐKXĐ: \(\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\)
\(A=\left(\frac{\left(\sqrt{a}\right)^2-1}{2\sqrt{a}}\right)^2\cdot\left(\frac{\left(\sqrt{a}-1\right)^2-\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right)\\ =\left(\frac{a-1}{2\sqrt{a}}\right)^2\cdot\left(\frac{a-2\sqrt{a}+1-a-2\sqrt{a}-1}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right)\\ =\frac{\left(a-1\right)^2}{4a}\cdot\frac{-4\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\\ =-\frac{a-1}{\sqrt{a}}=\frac{1-a}{\sqrt{a}}\)
b. Để A < 0 thì 1 - a <0 ( vì mẫu \(\sqrt{a}\ge0\forall a\) ) <=> -a < -1 <=> a > 1
\(A=\left(\frac{1}{\sqrt{a}-1}-\frac{1}{\sqrt{a}}\right):\left(\frac{\sqrt{a}+1}{\sqrt{a}-2}-\frac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)
\(=\frac{\sqrt{a}-\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\sqrt{a}}:\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)-\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)
\(=\frac{1}{\left(\sqrt{a}-1\right)\sqrt{a}}:\frac{a-1-\left(a-4\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)
\(=\frac{1}{\left(\sqrt{a}-1\right)\sqrt{a}}:\frac{3}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)
\(=\frac{1}{\left(\sqrt{a}-1\right)\sqrt{a}}.\frac{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}{3}\)
\(=\frac{\sqrt{a}-2}{3\sqrt{a}}\)
a) \(Q=\frac{a+2\sqrt{a}+1}{a-1}.\left(\frac{1}{\sqrt{a}-1}-\frac{2\sqrt{a}}{a\sqrt{a}-a+\sqrt{a}-1}\right)=\frac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}.\left[\frac{a+1}{\left(\sqrt{a}-1\right)\left(a+1\right)}-\frac{2\sqrt{a}}{\left(\sqrt{a}-1\right)\left(a+1\right)}\right]=\frac{\sqrt{a}+1}{\sqrt{a}-1}.\frac{a-2\sqrt{a}+1}{\left(\sqrt{a}-1\right)\left(a+1\right)}=\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)^2}{\left(\sqrt{a}-1\right)^2\left(a+1\right)}=\frac{\sqrt{a}+1}{a+1}\)
b) Ta có \(a>1\Leftrightarrow\sqrt{a}>1\Leftrightarrow\sqrt{a}-1>0\Leftrightarrow\sqrt{a}\left(\sqrt{a}-1\right)>0\Leftrightarrow a-\sqrt{a}>0\Leftrightarrow a+1>\sqrt{a}+1\Leftrightarrow\frac{\sqrt{a}+1}{a+1}< 1\Leftrightarrow Q< 1\)Vậy a>1 thì Q<1