Страница 93 номер 337, ГДЗ по алгебре за 9 класс к учебнику Мерзляка
Сократите дробь: 1) \(\frac{9 + \sqrt{m}}{m - 81}\); 2) \(\frac{\sqrt{27} + \sqrt{45}}{\sqrt{18} + \sqrt{30}}\); 3) \(\frac{\sqrt{5m} + \sqrt{7n}}{5m + 2\sqrt{35mn} + 7n}\); 4) \(\frac{25m + 10n\sqrt{3m} + 3n^2}{5\sqrt{m} + n\sqrt{3}}\).
1)
\(\dfrac{9 + \sqrt{m}}{m - 81} = \dfrac{9 + \sqrt{m}}{\left( \sqrt{m} - 9 \right)(\sqrt{m} + 9)} = \dfrac{1}{\sqrt{m} - 9}\)
2)
\(\dfrac{\sqrt{27} + \sqrt{45}}{\sqrt{18} + \sqrt{30}} = \dfrac{\sqrt{3 \cdot 9} + \sqrt{5 \cdot 9}}{\sqrt{3 \cdot 6} + \sqrt{6 \cdot 5}} = \dfrac{\sqrt{9}(\sqrt{3} + \sqrt{5})}{\sqrt{6}(\sqrt{3} + \sqrt{5})} = \dfrac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{2} \cdot \sqrt{3}} = \sqrt{\dfrac{3}{2}}\)
3)
\(\dfrac{\sqrt{5m} + \sqrt{7n}}{5m + 2\sqrt{35mn} + 7n} = \dfrac{\sqrt{5m} + \sqrt{7n}}{\left( \sqrt{5m} + \sqrt{7n} \right)^{2}} = \dfrac{1}{\sqrt{5m} + \sqrt{7n}}\)
4)
\(\dfrac{25m + 10n\sqrt{3m} + 3n^{2}}{5\sqrt{m} + n\sqrt{3}} = \dfrac{\left( 5\sqrt{m} + \sqrt{3}n \right)^{2}}{5\sqrt{m} + n\sqrt{3}} = 5\sqrt{m} + n\sqrt{3}\)
Ответ: 1) \(\frac{1}{\sqrt{m} - 9}\); 2) \(\sqrt{\frac{3}{2}}\); 3) \(\frac{1}{\sqrt{5m} + \sqrt{7n}}\); 4) \(5\sqrt{m} + n\sqrt{3}\).
