Упр.210(стар учебник) ГДЗ Дорофеев Суворова 8 класс (Алгебра)

Задача

Упростите выражение:

а) $$\frac{u-\frac{u^2}{u+1}}{u-\frac{u}{u+1}}$$;

б) $$\frac{x-\frac{6x-9}{x}}{\frac{3}{x}-1}$$;

в) $$\frac{\frac{a-b}{a+b}+\frac{b}{a}}{\frac{a}{a+b}+\frac{a-b}{a}}$$;

г) $$\frac{\frac{x}{x-z}-\frac{y}{y-z}}{\frac{y}{x-z}-\frac{x}{y-z}}$$;

д) $$1-\frac{1}{1-\frac{1}{n-3}}$$;

е) $$\frac{n}{n-\frac{1}{1-\frac{n}{1+n}}}$$.

Подробный ответ

а)
$$\frac{u-\frac{u^2}{u+1}}{u-\frac{u}{u+1}}= \frac{\frac{u(u+1)-u^2}{u+1}}{\frac{u(u+1)-u}{u+1}}= \frac{u}{u^2}=\frac{1}{u}.$$
б)
$$\frac{x-\frac{6x-9}{x}}{\frac{3}{x}-1}= \frac{\frac{x^2-6x+9}{x}}{\frac{3-x}{x}}= \frac{(x-3)^2}{3-x}=3-x.$$
в)
$$\frac{\frac{a-b}{a+b}+\frac{b}{a}}{\frac{a}{a+b}+\frac{a-b}{a}}= \frac{a(a-b)+b(a+b)}{a^2+(a-b)(a+b)}= \frac{a^2-ab+ab+b^2}{a^2+a^2-b^2}= \frac{a^2+b^2}{2a^2-b^2}.$$
г)
$$\frac{\frac{x}{x-z}-\frac{y}{y-z}}{\frac{y}{x-z}-\frac{x}{y-z}}= \frac{\left(\frac{x}{x-z}-\frac{y}{y-z}\right)(x-z)(y-z)}{\left(\frac{y}{x-z}-\frac{x}{y-z}\right)(x-z)(y-z)}$$
$$= \frac{x(y-z)-y(x-z)}{y(y-z)-x(x-z)}= \frac{xy-xz-xy+yz}{y^2-yz-x^2+xz}= \frac{z(y-x)}{(y-x)(y+x-z)}= \frac{z}{y+x-z}.$$
д)
$$1-\frac{1}{1-\frac{1}{n-3}}= 1-\frac{1}{\frac{n-3-1}{n-3}}= 1-\frac{n-3}{n-4}= \frac{n-4-n+3}{n-4}= -\frac{1}{n-4}= \frac{1}{4-n}.$$
е)
$$\frac{n}{\,n-\frac{1}{1-\frac{n}{1+n}}\,}= \frac{n}{\,n-\frac{1}{\frac{1+n-n}{1+n}}\,}= \frac{n}{\,n-(1+n)\,}= \frac{n}{-1}=-n.$$