$\frac{x-25}{\sqrt{x}-5}=\frac{(\sqrt{x}{)}^2-5^2}{\sqrt{x}-5}=\frac{(\sqrt{x}-5)(\sqrt{x}+5)}{\sqrt{x}-5}=\sqrt{x}+5$

$\frac{\sqrt{a}+2}{a-4}=\frac{\sqrt{a}+2}{(\sqrt{a}{)}^2-2^2}=\frac{\sqrt{a}+2}{(\sqrt{a}-2)(\sqrt{a}+2)}=\frac{1}{\sqrt{a}-2}$

$\frac{a-3}{\sqrt{a}+\sqrt{3}}=\frac{(\sqrt{a}{)}^2-(\sqrt{3}{)}^2}{\sqrt{a}+\sqrt{3}}=\frac{(\sqrt{a}-\sqrt{3})(\sqrt{a}+\sqrt{3})}{\sqrt{a}+\sqrt{3}}=\sqrt{a}-\sqrt{3}$

$\frac{\sqrt{10}+\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{2\ast 5}+\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{2}\ast \sqrt{5}+\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}(\sqrt{2}+1)}{\sqrt{5}}=\sqrt{2}+1$

$\frac{23-\sqrt{23}}{\sqrt{23}}=\frac{(\sqrt{23}{)}^2-\sqrt{23}}{\sqrt{23}}=\frac{\sqrt{23}(\sqrt{23}-1)}{\sqrt{23}}=\sqrt{23}-1$

$\frac{\sqrt{24}-\sqrt{28}}{\sqrt{54}-\sqrt{63}}=\frac{\sqrt{4\ast 6}-\sqrt{4\ast 7}}{\sqrt{9\ast 6}-\sqrt{9\ast 7}}=\frac{2\sqrt{6}-2\sqrt{7}}{3\sqrt{6}-3\sqrt{7}}=\frac{2(\sqrt{6}-\sqrt{7})}{3(\sqrt{6}-\sqrt{7})}=\frac{2}{3}$

$\frac{\sqrt{a}-\sqrt{b}}{a-2\sqrt{ab}+b}=\frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}{)}^2-2\ast \sqrt{a}\ast \sqrt{b}+(\sqrt{b}{)}^2}=\frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}-\sqrt{b}{)}^2}=\frac{1}{\sqrt{a}-\sqrt{b}}$

$\frac{b-8\sqrt{b}+16}{\sqrt{b}-4}=\frac{(\sqrt{b}{)}^2-2\ast \sqrt{b}\ast 4+4^2}{\sqrt{b}-4}=\frac{(\sqrt{b}-4{)}^2}{\sqrt{b}-4}=\sqrt{b}-4$