a: \(P=\frac{\sqrt{x}+2}{\sqrt{x}-1}-\frac{6}{\sqrt{x}+1}+\frac{x-7}{x-1}\)
\(=\frac{\left(\sqrt{x}+2\right)\cdot\left(\sqrt{x}+1\right)-6\left(\sqrt{x}-1\right)+x-7}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+3\sqrt{x}+2-6\sqrt{x}+6+x-7}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{2x-3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)\left(2\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{2\sqrt{x}-1}{\sqrt{x}+1}\)
b: \(x=\left(\sqrt7+\sqrt3\right)\cdot\sqrt{10-2\sqrt{21}}\)
\(=\left(\sqrt7+\sqrt3\right)\cdot\sqrt{\left(\sqrt7-\sqrt3\right)^2}\)
\(=\left(\sqrt7+\sqrt3\right)\left(\sqrt7-\sqrt3\right)=7-3=4\)
Thay x=4 vào P, ta được:
\(P=\frac{2\cdot2-1}{2+1}=\frac33=1\)
