Упр.256 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) ab^-1 + a^-1 * b;
2) 3a^-1 + ab^-2;
3) m2n2(m^-3 — n^-3);
4) (a+b)^-1 * (a^-1 + b^-1);
5) (c^-2 — d^-2) : (c+d);
6) (xy^-2 + x^-2 y) * ((x2-xy+y2)/x)^-1.

1. $$ab^{-1}+a^{-1}b=\frac{a}{b}+\frac{b}{a}=\frac{a^2+b^2}{ab}.$$
2. $$3a^{-1}+ab^{-2}=\frac{3}{a}+\frac{a}{b^2}=\frac{3b^2+a^2}{ab^2}.$$
3. $$ m^2n^2(m^{-3}-n^{-3})=m^2n^2\left(\frac{1}{m^3}-\frac{1}{n^3}\right) =\frac{n^2}{m}-\frac{m^2}{n} =\frac{n^3-m^3}{mn}. $$
4. $$ (a+b)^{-1}(a^{-1}+b^{-1}) =\frac{1}{a+b}\left(\frac{1}{a}+\frac{1}{b}\right) =\frac{1}{a+b}\cdot\frac{a+b}{ab} =\frac{1}{ab}. $$
5. $$ (c^{-2}-d^{-2}):(c+d) =\left(\frac{1}{c^2}-\frac{1}{d^2}\right): (c+d) =\frac{d^2-c^2}{c^2d^2}\cdot\frac{1}{c+d} =\frac{(d-c)(d+c)}{c^2d^2}\cdot\frac{1}{c+d} =\frac{d-c}{c^2d^2}. $$
6. $$ \left(xy^{-2}+x^{-2}y\right)\left(\frac{x^2-xy+y^2}{x}\right)^{-1} =\left(\frac{x}{y^2}+\frac{y}{x^2}\right)\cdot\frac{x}{x^2-xy+y^2} $$
   $$ =\frac{x^3+y^3}{x^2y^2}\cdot\frac{x}{x^2-xy+y^2} =\frac{(x+y)(x^2-xy+y^2)}{x^2y^2}\cdot\frac{x}{x^2-xy+y^2} =\frac{x+y}{xy^2}. $$

Ответ

1) $$\frac{a^2+b^2}{ab}$$; 2) $$\frac{3b^2+a^2}{ab^2}$$; 3) $$\frac{n^3-m^3}{mn}$$; 4) $$\frac{1}{ab}$$; 5) $$\frac{d-c}{c^2d^2}$$; 6) $$\frac{x+y}{xy^2}$$.