\[\boxed{\text{410.}\text{\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
Пояснение.
Решение.
\[\sqrt{a + \frac{a}{b}} = a\sqrt{\frac{a}{b}}\]
\[1.\ \sqrt{2\frac{2}{3}} = 2\sqrt{\frac{2}{3}}\text{\ \ \ \ \ \ \ \ \ \ \ }\sqrt{3\frac{3}{8}} = 3\sqrt{\frac{3}{8}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\]
\[\text{\ \ \ }\sqrt{4\frac{4}{15}} = 4\sqrt{\frac{4}{15}}\text{\ \ \ \ \ \ \ }\sqrt{\frac{8}{3}} = \sqrt{\frac{4 \cdot 2}{3}}\text{\ \ \ \ \ \ \ }\]
\[\ \sqrt{\frac{24 + 3}{8}} = \sqrt{\frac{9 \cdot 3}{8}}\text{\ \ \ \ \ \ \ \ \ }\]
\[\text{\ \ }\sqrt{\frac{60 + 4}{15}} = \sqrt{\frac{16 \cdot 4}{15}}\text{\ \ \ \ \ }\]
\[\text{\ \ }\sqrt{\frac{8}{3}} = \sqrt{\frac{8}{3}}\text{\ \ \ \ \ \ \ \ \ }\sqrt{\frac{27}{8}} = \sqrt{\frac{27}{8}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ }\]
\[\ \sqrt{\frac{64}{15}} = \sqrt{\frac{64}{15}}\]
\[2.\ \left( \sqrt{a + \frac{a}{b}} \right)^{2} = \left( a\sqrt{\frac{a}{b}} \right)^{2}\]
\[a + \frac{a}{b} = a^{2} \cdot \frac{a}{b}\]
\[\frac{ab + a}{b} = \frac{a^{3}}{b}\]
\[a^{2} = b + 1\]
\[b = a^{2} - 1\]
\[3.\ \]
\[b = 3^{2} - 1 = - 1 = 8\ \ \ \ \ \ \ \ \ \ \ \]
\[\ \sqrt{3\frac{3}{8}} = 3\sqrt{\frac{3}{8}}.\]
\[b = 5^{2} - 1 = 25 - 1 = 24\ \ \ \ \]
\[\ \sqrt{5\frac{5}{24}} = 5\sqrt{\frac{5}{24}}.\]