a: \(A=\sqrt{5+2\sqrt6}-\sqrt3\)
\(=\sqrt{\left(\sqrt3+\sqrt2\right)^2}-\sqrt3\)
\(=\sqrt3+\sqrt2-\sqrt3=\sqrt2\)
b: \(B=\sqrt{8-2\sqrt{15}}+\sqrt5+\sqrt3\)
\(=\sqrt{\left(\sqrt5-\sqrt3\right)^2}+\sqrt5+\sqrt3\)
\(=\sqrt5-\sqrt3+\sqrt5+\sqrt3=2\sqrt5\)
c: \(C=\sqrt{7+2\sqrt{10}}-\sqrt{7-2\sqrt{10}}\)
\(=\sqrt{\left(\sqrt5+\sqrt2\right)^2}-\sqrt{\left(\sqrt5-\sqrt2\right)^2}\)
\(=\sqrt5+\sqrt2-\left(\sqrt5-\sqrt2\right)=2\sqrt2\)
d: \(D=\sqrt{9-2\sqrt{14}}-\sqrt{9+2\sqrt{14}}\)
\(=\sqrt{\left(\sqrt7-\sqrt2\right)^2}-\sqrt{\left(\sqrt7+\sqrt2\right)^2}\)
\(=\sqrt7-\sqrt2-\sqrt7-\sqrt2=-2\sqrt2\)
e: \(E=\frac{\sqrt{24}-\sqrt{40}}{\sqrt{54}-\sqrt{90}}\)
\(=\frac{2\sqrt6-2\sqrt{10}}{3\sqrt6-3\sqrt{10}}\)
\(=\frac{2\left(\sqrt6-\sqrt{10}\right)}{3\left(\sqrt6-\sqrt{10}\right)}=\frac23\)
f: \(M=\frac{a+\sqrt{ab}}{b+\sqrt{ab}}\)
\(=\frac{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)}=\frac{\sqrt{a}}{\sqrt{b}}\)
g: \(N=\frac{1+\sqrt{x}+\sqrt{y}+\sqrt{xy}}{1+\sqrt{y}}\)
\(=\frac{\left(1+\sqrt{y}\right)+\sqrt{x}\left(1+\sqrt{y}\right)}{1+\sqrt{y}}\)
\(=\frac{\left(1+\sqrt{y}\right)\left(1+\sqrt{x}\right)}{1+\sqrt{y}}=1+\sqrt{x}\)
i: \(P=\frac{4y+3\sqrt{y}-7}{4\sqrt{y}+7}\)
\(=\frac{4y+7\sqrt{y}-4\sqrt{y}-7}{4\sqrt{y}+7}\)
\(=\frac{\left(4\sqrt{y}+7\right)\left(\sqrt{y}-1\right)}{4\sqrt{y}+7}=\sqrt{y}-1\)
p: \(Q=\frac{x-3\sqrt{x}-4}{x-\sqrt{x}-12}\)
\(=\frac{x-4\sqrt{x}+\sqrt{x}-4}{x-4\sqrt{x}+3\sqrt{x}-12}\)
\(=\frac{\left.\left(\sqrt{x}-4\right)\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-4\right)\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}+1}{\sqrt{x}+3}\)
