Первый пример
${\left(\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{b}}{\sqrt{a}}+2\right)\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=}$
${=\left(\frac{a+b+2\sqrt{ab}}{\sqrt{ab}}\right)\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{a+2\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{\left(\sqrt{a}+\sqrt{b}\right)^{2}}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}+\sqrt{b}}=}$
${=\sqrt{a}+\sqrt{b}}$
Второй пример
${\sqrt{9-4\sqrt{5}}=\sqrt{(5+4)-4\sqrt{5}}=}$
${=\sqrt{5-4\sqrt{5}+4}=\sqrt{\sqrt{5}^2-2*2\sqrt{5}+2^2}=}$
${=\sqrt{\left(\sqrt{5}-2\right)^2}=\sqrt{5}-2}$
${\left(\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{b}}{\sqrt{a}}+2\right)\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=}$
${=\left(\frac{a+b+2\sqrt{ab}}{\sqrt{ab}}\right)\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{a+2\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{\left(\sqrt{a}+\sqrt{b}\right)^{2}}{\sqrt{a}+\sqrt{b}}=}$
${=\frac{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}+\sqrt{b}}=}$
${=\sqrt{a}+\sqrt{b}}$
Второй пример
${\sqrt{9-4\sqrt{5}}=\sqrt{(5+4)-4\sqrt{5}}=}$
${=\sqrt{5-4\sqrt{5}+4}=\sqrt{\sqrt{5}^2-2*2\sqrt{5}+2^2}=}$
${=\sqrt{\left(\sqrt{5}-2\right)^2}=\sqrt{5}-2}$
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