\(đk:x\ge\frac{-3}{2}\)

\(\sqrt{8x+13+4\sqrt{2x+3}}+\sqrt{2x+7-4\sqrt{2x+3}}=9\)

\(\Leftrightarrow\sqrt{4\left(2x+3\right)+4\sqrt{2x+3}+1}+\sqrt{2x+3-4\sqrt{2x+3}+4}=9\)

\(\Leftrightarrow\sqrt{\left(2\sqrt{2x+3}+1\right)^2}+\sqrt{\left(\sqrt{2x+3}+2\right)^2}=9\)

\(\Leftrightarrow|2\sqrt{2x+3}+1|+|\sqrt{2x+3}+2|=9\Leftrightarrow3\sqrt{2x+3}=6\Leftrightarrow\sqrt{2x+3}=2\Leftrightarrow2x+3=4\)

\(\Leftrightarrow x=\frac{1}{2}\left(\text{thỏa mãn}\right)\)

đkxđ: ....

\(\sqrt{x+4}+\sqrt{x+11}=x+27-x^2\)

\(\Leftrightarrow x+4+2\sqrt{\left(x+4\right)\left(x+11\right)}+x+1=x^2+729+x^4+54x-2x^3-54x^2\)

\(\Leftrightarrow2x+5+2\sqrt{\left(x+4\right)\left(x+11\right)}=x^4-2x^3-53x^2+54x+729\)

\(\Leftrightarrow2\sqrt{x^2+15x+44}=x^4-2x^3-53x^2+52x+724\)

\(\Leftrightarrow2\sqrt{x^2+15x+44}=\left(x-2\right)\left(x^3-53x-54\right)+616\) 

.........

\(\left(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\sqrt{xy}\right)\cdot\frac{1}{x-y}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=\left(\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-\sqrt{xy}\right)\cdot\frac{1}{x-y}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=\left(x-\sqrt{xy}+y-\sqrt{xy}\right)\cdot\frac{1}{x-y}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=\left(x-2\sqrt{xy}+y\right)\cdot\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

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

\(=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}+\frac{2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=\frac{\sqrt{x}-\sqrt{y}+2\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)

\(=1\)