Страница 33 номер 1.100, ГДЗ по алгебре за 8 класс к учебнику Дорофеева

Преобразование каждого выражения выполняем по действиям.

а)

1) \(m+3+\frac{9}{m-3} = \frac{(m+3)(m-3)+9}{m-3} = \frac{m^2}{m-3}\);

2) \(\frac{m}{m-3}+\frac{3m}{(3-m)^2} = \frac{m}{m-3}+\frac{3m}{(m-3)^2} = \frac{m(m-3)+3m}{(m-3)^2} = \frac{m^2}{(m-3)^2}\);

3) \(\frac{m^2}{m-3}:\frac{m^2}{(m-3)^2} = \frac{m^2(m-3)^2}{(m-3)m^2} = m-3\).

б)

1) \(\frac{n}{1+2n+n^2}-\frac{n}{n+1} = \frac{n}{(n+1)^2}-\frac{n(n+1)}{(n+1)^2} = \frac{n-n^2-n}{(n+1)^2} = \frac{-n^2}{(n+1)^2}\);

2) \(\frac{1}{n+1}+n-1 = \frac{1+(n-1)(n+1)}{n+1} = \frac{n^2}{n+1}\);

3) \(\frac{-n^2}{(n+1)^2}:\frac{n^2}{n+1} = \frac{-n^2(n+1)}{(n+1)^2n^2} = -\frac{1}{n+1}\).

в)

1) \(\frac{x^3}{y^3}+1 = \frac{x^3+y^3}{y^3} = \frac{(x+y)(x^2-xy+y^2)}{y^3}\);

2) \(\frac{x}{y^2}-\frac{1}{y}+\frac{1}{x} = \frac{x^2-xy+y^2}{xy^2}\);

3) \(\frac{(x+y)(x^2-xy+y^2)}{y^3}:\frac{x^2-xy+y^2}{xy^2} = \frac{(x+y)(x^2-xy+y^2)xy^2}{y^3(x^2-xy+y^2)} = \frac{x^2+xy}{y}\).

г)

1) \(1+\frac{v}{u}+\frac{v^2}{u^2} = \frac{u^2+uv+v^2}{u^2}\);

2) \(\frac{1}{v}-\frac{v^2}{u^3} = \frac{u^3-v^3}{u^3v} = \frac{(u-v)(u^2+uv+v^2)}{u^3v}\);

3) \(\frac{u^2+uv+v^2}{u^2}:\frac{(u-v)(u^2+uv+v^2)}{u^3v} = \frac{(u^2+uv+v^2)u^3v}{u^2(u-v)(u^2+uv+v^2)} = \frac{uv}{u-v}\).

Ответ: а) \(m-3\); б) \(-\frac{1}{n+1}\); в) \(\frac{x^2+xy}{y}\); г) \(\frac{uv}{u-v}\).