Упр.252 Повторение ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) (va-5)/v(a-1)-(va-4)/va;
2) (va+1)/(a+v(ab))-(vb-1)/(v(ab)+b);
3) ((vx+2)/(vx-2)-8vx/(x-4))·(x+2vx)/(vx-2);
4) (a-49)/(va+2)·1/(a+7va)-(va+7)/(a-2va).

1. $$\frac{\sqrt a-5}{\sqrt a-1}-\frac{\sqrt a-4}{\sqrt a}=
   \frac{\sqrt a(\sqrt a-5)-(\sqrt a-4)(\sqrt a-1)}{\sqrt a(\sqrt a-1)}$$

   $$=\frac{a-5\sqrt a-\left(a-5\sqrt a+4\right)}{\sqrt a(\sqrt a-1)}
   =\frac{-4}{\sqrt a(\sqrt a-1)}=\frac{4}{\sqrt a-a}$$

2. $$\frac{\sqrt a+1}{a+\sqrt{ab}}-\frac{\sqrt b-1}{\sqrt{ab}+b}
   =\frac{\sqrt a+1}{\sqrt a(\sqrt a+\sqrt b)}-\frac{\sqrt b-1}{\sqrt b(\sqrt a+\sqrt b)}$$

   $$=\frac{\sqrt b(\sqrt a+1)-\sqrt a(\sqrt b-1)}{\sqrt{ab}(\sqrt a+\sqrt b)}
   =\frac{\sqrt{ab}+\sqrt b-\sqrt{ab}+\sqrt a}{\sqrt{ab}(\sqrt a+\sqrt b)}
   =\frac{\sqrt a+\sqrt b}{\sqrt{ab}(\sqrt a+\sqrt b)}=\frac1{\sqrt{ab}}$$

3. $$\left(\frac{\sqrt x+2}{\sqrt x-2}-\frac{8\sqrt x}{x-4}\right)\cdot\frac{x+2\sqrt x}{\sqrt x-2}$$

   Так как $$x-4=(\sqrt x-2)(\sqrt x+2),$$ то

   $$\frac{\sqrt x+2}{\sqrt x-2}-\frac{8\sqrt x}{(\sqrt x-2)(\sqrt x+2)}
   =\frac{(\sqrt x+2)^2-8\sqrt x}{(\sqrt x-2)(\sqrt x+2)}$$

   $$=\frac{x+4\sqrt x+4-8\sqrt x}{(\sqrt x-2)(\sqrt x+2)}
   =\frac{x-4\sqrt x+4}{(\sqrt x-2)(\sqrt x+2)}
   =\frac{(\sqrt x-2)^2}{(\sqrt x-2)(\sqrt x+2)}$$

   Тогда
   $$\frac{(\sqrt x-2)^2}{(\sqrt x-2)(\sqrt x+2)}\cdot\frac{x+2\sqrt x}{\sqrt x-2}
   =\frac{\sqrt x(\sqrt x+2)}{\sqrt x+2}=\sqrt x.$$

4. $$\frac{a-49}{\sqrt a+2}\cdot\frac1{a+7\sqrt a}-\frac{\sqrt a+7}{a-2\sqrt a}$$

   Разложим на множители:
   $$a-49=(\sqrt a-7)(\sqrt a+7),\quad a+7\sqrt a=\sqrt a(\sqrt a+7),\quad a-2\sqrt a=\sqrt a(\sqrt a-2).$$

   Тогда
   $$\frac{(\sqrt a-7)(\sqrt a+7)}{(\sqrt a+2)\sqrt a(\sqrt a+7)}-\frac{\sqrt a+7}{\sqrt a(\sqrt a-2)}
   =\frac{\sqrt a-7}{\sqrt a(\sqrt a+2)}-\frac{\sqrt a+7}{\sqrt a(\sqrt a-2)}$$

   $$=\frac{(\sqrt a-7)(\sqrt a-2)-(\sqrt a+7)(\sqrt a+2)}{\sqrt a(\sqrt a+2)(\sqrt a-2)}$$

   $$=\frac{a-9\sqrt a+14-a-9\sqrt a-14}{\sqrt a(a-4)}
   =\frac{-18\sqrt a}{\sqrt a(a-4)}=\frac{-18}{a-4}=\frac{18}{4-a}.$$

Ответ

1) $$\frac{4}{\sqrt a-a}$$; 2) $$\frac1{\sqrt{ab}}$$; 3) $$\sqrt x$$; 4) $$\frac{18}{4-a}$$.