Упр.106 Вариант 1 Дидактические материалы ГДЗ Мерзляк Полонский 8 класс (Алгебра)

Упростите выражение:

1. $$\frac{a}{a-1}-\frac{\sqrt{a}}{\sqrt{a}+1}$$;

   $$\frac{a+b}{\sqrt{ab}-b}-\frac{2\sqrt{a}}{\sqrt{a}-\sqrt{b}}$$;

   $$\frac{\sqrt{x}-6}{\sqrt{x}}:\frac{x-36}{4x}$$;

   $$\left(\frac{\sqrt{a}-5}{\sqrt{a}+5}+\frac{20\sqrt{a}}{a-25}\right):\frac{\sqrt{a}+5}{a-5\sqrt{a}}$$.

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

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

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

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

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

   $$=\frac{(\sqrt a-\sqrt b)^2}{\sqrt b(\sqrt a-\sqrt b)}=\frac{\sqrt a-\sqrt b}{\sqrt b}.$$

3. $$\frac{\sqrt x-6}{\sqrt x}:\frac{x-36}{4x}=\frac{\sqrt x-6}{\sqrt x}\cdot\frac{4x}{x-36}$$

   $$=\frac{\sqrt x-6}{\sqrt x}\cdot\frac{4x}{(\sqrt x-6)(\sqrt x+6)}=\frac{4x}{\sqrt x(\sqrt x+6)}$$

   $$=\frac{4\sqrt x}{\sqrt x+6}.$$

4. $$\left(\frac{\sqrt a-5}{\sqrt a+5}+\frac{20\sqrt a}{a-25}\right):\frac{\sqrt a+5}{a-5\sqrt a}$$

   $$=\left(\frac{\sqrt a-5}{\sqrt a+5}+\frac{20\sqrt a}{(\sqrt a-5)(\sqrt a+5)}\right)\cdot\frac{a-5\sqrt a}{\sqrt a+5}$$

   $$=\frac{(\sqrt a-5)^2+20\sqrt a}{(\sqrt a-5)(\sqrt a+5)}\cdot\frac{\sqrt a(\sqrt a-5)}{\sqrt a+5}$$

   $$=\frac{a-10\sqrt a+25+20\sqrt a}{(\sqrt a-5)(\sqrt a+5)}\cdot\frac{\sqrt a(\sqrt a-5)}{\sqrt a+5}$$

   $$=\frac{a+10\sqrt a+25}{(\sqrt a-5)(\sqrt a+5)}\cdot\frac{\sqrt a(\sqrt a-5)}{\sqrt a+5}$$

   $$=\frac{(\sqrt a+5)^2}{(\sqrt a-5)(\sqrt a+5)}\cdot\frac{\sqrt a(\sqrt a-5)}{\sqrt a+5}=\sqrt a.$$

Ответ: 1) $$\frac{\sqrt a}{a-1}$$; 2) $$\frac{\sqrt a-\sqrt b}{\sqrt b}$$; 3) $$\frac{4\sqrt x}{\sqrt x+6}$$; 4) $$\sqrt a$$.