Страница 16 номер 58, ГДЗ по алгебре за 9 класс к учебнику Мерзляка
Упростите выражение: 1) \(\frac{a^2 + b^2}{2a^2 + 2ab} + \frac{b}{a + b}\); 2) \(\frac{a^2 + 9}{a^2 - 9} - \frac{a}{a + 3}\); 3) \(\frac{c + 1}{3c} : \frac{c^2 - 1}{6c^2}\); 4) \(\frac{m^2 + 2mn + n^2}{m^2 - n^2} : (m + n)\).
1) \(\frac{a^{2} + b^{2}}{2a^{2} + 2ab} + \frac{b}{a + b} = \frac{a^{2} + b^{2}}{2a(a + b)} + \frac{2a \cdot b}{2a(a + b)} = \frac{a^{2} + 2ab + b^{2}}{2a(a + b)} = \frac{(a + b)^{2}}{2a(a + b)} = \frac{a + b}{2a}\)
2) \(\frac{a^{2} + 9}{a^{2} - 9} - \frac{a}{a + 3} = \frac{a^{2} + 9}{a^{2} - 9} - \frac{a(a - 3)}{(a + 3)(a - 3)} = \frac{a^{2} + 9 - a^{2} + 3a}{(a + 3)(a - 3)} = \frac{3(a + 3)}{(a + 3)(a - 3)} = \frac{3}{a - 3}\)
3) \(\frac{c + 1}{3c}:\frac{c^{2} - 1}{6c^{2}} = \frac{c + 1}{3c} \cdot \frac{6c^{2}}{c^{2} - 1} = \frac{c + 1}{3c} \cdot \frac{3c \cdot 2c}{(c - 1)(c + 1)} = \frac{2c}{c - 1}\)
4) \(\frac{m^{2} + 2mn + n^{2}}{m^{2} - n^{2}}:(m + n) = \frac{(m + n)^{2}}{(m - n)(m + n)} \cdot \frac{1}{m + n} = \frac{1}{m - n}\)
Ответ: 1) \(\frac{a + b}{2a}\); 2) \(\frac{3}{a - 3}\); 3) \(\frac{2c}{c - 1}\); 4) \(\frac{1}{m - n}\).
