Упр.1.88 ГДЗ Дорофеев Суворова 8 класс (Алгебра)

а) (x^3+y^3)/(x^3-y^3 )•(y-x)/(y+x);
б) (2x^3+2z^3)/(xz-x^2 ) :(x^3-x^2 z+xz^2)/(x^2-z^2 );
в) (p^3-q^3)/(p^4-q^4 )•(p^2+q^2)/(p-q);
г) (x^4-16)/(x^2-4x+4) :(x^3+8)/(x-2).

1. а)

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

2. б)

   $$\frac{2x^3+2z^3}{xz-x^2}:\frac{x^3-x^2z+xz^2}{x^2-z^2} =\frac{2(x^3+z^3)}{x(z-x)}\cdot\frac{x^2-z^2}{x^3-x^2z+xz^2}$$
   $$=\frac{2(x+z)(x^2-xz+z^2)}{-x(x-z)}\cdot\frac{(x-z)(x+z)}{x(x^2-xz+z^2)}$$
   $$=-\frac{2(x+z)^2}{x^2}.$$

3. в)

   $$\frac{p^3-q^3}{p^4-q^4}\cdot\frac{p^2+q^2}{p-q} =\frac{(p-q)(p^2+pq+q^2)}{(p^2-q^2)(p^2+q^2)}\cdot\frac{p^2+q^2}{p-q}$$
   $$=\frac{p^2+pq+q^2}{p^2-q^2}.$$

4. г)

   $$\frac{x^4-16}{x^2-4x+4}:\frac{x^3+8}{x-2} =\frac{(x^2-4)(x^2+4)}{(x-2)^2}\cdot\frac{x-2}{x^3+8}$$
   $$=\frac{(x-2)(x+2)(x^2+4)}{(x-2)^2}\cdot\frac{x-2}{(x+2)(x^2-2x+4)}$$
   $$=\frac{x^2+4}{x^2-2x+4}.$$

Ответ

а) $$\frac{xy-x^2-y^2}{x^2+xy+y^2}$$;
б) $$-\frac{2(x+z)^2}{x^2}$$;
в) $$\frac{p^2+pq+q^2}{p^2-q^2}$$;
г) $$\frac{x^2+4}{x^2-2x+4}$$.