Упр.8.20 ГДЗ Мордкович 8 класс (Алгебра)

1. а) $$\left(b^{-1}+a^{-1}\right)\cdot(a+b)^{-1}$$;
   б) $$\left(x^{-2}-y^{-2}\right):(x-y)$$;
   в) $$\left(m^{-2}+n^{-2}\right):(m^2+n^2)$$;
   г) $$\left(ab^{-2}+a^{-2}b\right)\cdot\left(\frac{a^{-1}}{b}\right)^{-2}$$.

1. $$(b^{-1}+a^{-1})\cdot(a+b)^{-1} =\left(\frac{1}{b}+\frac{1}{a}\right)\cdot\frac{1}{a+b} =\frac{a+b}{ab}\cdot\frac{1}{a+b} =\frac{1}{ab}.$$

2. $$(x^{-2}-y^{-2}):(x-y) =\left(\frac{1}{x^2}-\frac{1}{y^2}\right):(x-y) =\frac{y^2-x^2}{x^2y^2}:(x-y) =-\frac{(x-y)(x+y)}{x^2y^2}:(x-y) =-\frac{x+y}{x^2y^2}.$$

3. $$(m^{-2}+n^{-2}):(m^2+n^2) =\left(\frac{1}{m^2}+\frac{1}{n^2}\right):(m^2+n^2) =\frac{m^2+n^2}{m^2n^2}:\,(m^2+n^2) =\frac{1}{m^2n^2}.$$

4. $$(ab^{-2}+a^{-2}b)\cdot\left(\frac{a^{-1}}{b}\right)^{-2} =\left(\frac{a}{b^2}+\frac{b}{a^2}\right)\cdot\left(\frac{1}{ab}\right)^{-2} =\frac{a^3+b^3}{a^2b^2}\cdot a^2b^2 =a^3+b^3.$$

Ответ

а) $$\frac{1}{ab}$$; б) $$-\frac{x+y}{x^2y^2}$$; в) $$\frac{1}{m^2n^2}$$; г) $$a^3+b^3$$.