Упр.40.26 ГДЗ Мерзляк Поляков 7 класс (Алгебра)

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

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) $$\frac{x+y}{xy^2}$$.