ta có :

\(\left(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\right)^2=4+\sqrt{7}+4-\sqrt{7}-2\sqrt{4+\sqrt{7}}.\sqrt{4-\sqrt{7}}\)

\(=8-2\sqrt{16-7}=8-2\sqrt{9}=2\)

vậy \(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}=\sqrt{2}\)

\(\sqrt{4-\sqrt{15}}\cdot\left(4+\sqrt{15}\right)\cdot\left(\sqrt{10}-\sqrt{6}\right)=\left(4+\sqrt{15}\right)\sqrt{8-2\sqrt{15}.}\left(\sqrt{5}-\sqrt{3}\right)\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)^2=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)=2>\sqrt{3}\)

vậy số bên trái lơn hơn \(\sqrt{3}\)

\(\sqrt{11-4\sqrt{7}}-\sqrt{29-4\sqrt{7}}\)

\(=\sqrt{7-2.\sqrt{7}.2+4}-\sqrt{28-2.2\sqrt{7}.1+1}\)

\(=\sqrt{\left(\sqrt{7}\right)^2-2.\sqrt{7}.2+2^2}-\sqrt{\left(2\sqrt{7}\right)^2-2.2\sqrt{7}.1+1^2}\)

\(=\sqrt{\left(\sqrt{7}-2\right)^2}-\sqrt{\left(2\sqrt{7}-1\right)^2}\)

\(=\left|\sqrt{7}-2\right|-\left|2\sqrt{7}-1\right|\)

\(=\sqrt{7}-2-2\sqrt{7}+1\)

\(=-\sqrt{7}-1\)

Bài 2. 

ĐKXĐ của biểu thức đã cho là: 

\(\hept{\begin{cases}x\ge0,\sqrt{x}\ne0\\\sqrt{x}-1\ne0\\\sqrt{x}-2\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x>0\\x\ne1,x\ne2\end{cases}}\).

\(A=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right)\div\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)

\(=\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}\div\left[\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right]\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\div\left(\frac{x-1-x+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)=\frac{\sqrt{x}-2}{\sqrt{x}}\)

\(A>\frac{1}{6}\Rightarrow\frac{\sqrt{x}-2}{\sqrt{x}}>\frac{1}{6}\Leftrightarrow6\left(\sqrt{x}-2\right)>\sqrt{x}\)

\(\Leftrightarrow5\sqrt{x}>12\Leftrightarrow x>\frac{144}{25}\).

ĐK: \(-\sqrt{3}\le x\le\sqrt{3}\).

\(\left(x-2\right)\sqrt{3-x^2}=x^2-x-2\)

\(\Leftrightarrow\left(x-2\right)\sqrt{3-x^2}=\left(x-2\right)\left(x+1\right)\)

\(\Leftrightarrow\left(x-2\right)\left(\sqrt{3-x^2}-x-1\right)=0\)

\(\Leftrightarrow\sqrt{3-x^2}=x+1\)(vì \(-\sqrt{3}\le x\le\sqrt{3}\))

\(\Rightarrow3-x^2=\left(x+1\right)^2\)

\(\Leftrightarrow2x^2+2x-2=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{-1+\sqrt{5}}{2}\left(tm\right)\\x=\frac{-1-\sqrt{5}}{2}\left(l\right)\end{cases}}\)