Монгол Бодлогын Сан

Заавар: $\sqrt{2}+\sqrt{5}-\sqrt{7}$ тоогоор хүртвэр ба хуваарийг үржүүл.

Бодолт: $$\begin{align*} \alpha&=\dfrac{1}{\sqrt{2}+\sqrt{5}+\sqrt{7}}\\ &=\dfrac{\sqrt{2}+\sqrt{5}-\sqrt{7}}{(\sqrt{2}+\sqrt{5}+\sqrt{7})(\sqrt{2}+\sqrt{5}-\sqrt{7})}\\ &=\dfrac{\sqrt{2}+\sqrt{5}-\sqrt{7}}{(\sqrt{2}+\sqrt{5})^2-(\sqrt{7^2})}\\ &=\dfrac{\sqrt{2}+\sqrt{5}-\sqrt{7}}{2+2\sqrt{10}+5-7}\\ &=\dfrac{\sqrt{2}+\sqrt{5}-\sqrt{7}}{2\sqrt{10}} & & \color{WildStrawberry}{\leftarrow\dfrac{{}\times\sqrt{10}}{{}\times\sqrt{10}}} \\ &=\dfrac{5\sqrt{2}+2\sqrt{5}-\sqrt{70}}{20} \end{align*}$$