е) $\dfrac{\sqrt{5}}{\sqrt{5}-\sqrt{2}}+\dfrac{\sqrt{5}}{\sqrt{5}+\sqrt{2}}$.

Question

Вопрос: $Фото задания: е) $\dfrac{\sqrt{5}}{\sqrt{5}-\sqrt{2}}+\dfrac{\sqrt{5}}{\sqrt{5}+\sqrt{2}}$.$ Фото задания

Accepted Answer

Рационализируем знаменатели:

\[\frac{\sqrt{5}}{\sqrt{5}-\sqrt{2}}=\frac{\sqrt{5}(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}=\frac{5+\sqrt{10}}{3}.\]

\[\frac{\sqrt{5}}{\sqrt{5}+\sqrt{2}}=\frac{\sqrt{5}(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}=\frac{5-\sqrt{10}}{3}.\]

Складываем:

\[\frac{5+\sqrt{10}}{3}+\frac{5-\sqrt{10}}{3}=\frac{10}{3}.\]

Ответ: $\frac{10}{3}$.

е) \(\dfrac{\sqrt{5}}{\sqrt{5}-\sqrt{2}}+\dfrac{\sqrt{5}}{\sqrt{5}+\sqrt{2}}\).