Упр.17.29 ГДЗ Мерзляк Поляков 8 класс (Алгебра)

1) (va-3)/(va+1)-(va-4)/va;
2) (va+1)/(a-vab)-(vb+1)/(vab-b);
3) vx/(y-2vy) :vx/(3vy-6);
4) vm/(vm-vn) :((vm+vn)/vn+vn/(vm-vn));
5) ((vx+1)/(vx-1)-(4vx)/(x-1))•(x+vx)/(vx-1);
6) (a-64)/(va+3)•1/(a+8va)-(va+8)/(a-3va).

1. $$\frac{\sqrt a-3}{\sqrt a+1}-\frac{\sqrt a-4}{\sqrt a}= \frac{\sqrt a(\sqrt a-3)-(\sqrt a-4)(\sqrt a+1)}{\sqrt a(\sqrt a+1)}$$

   $$=\frac{a-3\sqrt a-(a+\sqrt a-4\sqrt a-4)}{\sqrt a(\sqrt a+1)}= \frac{4}{\sqrt a(\sqrt a+1)}=\frac{4}{a+\sqrt a}$$

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

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

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

3. $$\frac{\sqrt x}{y-2\sqrt y}:\frac{\sqrt x}{3\sqrt y-6}= \frac{\sqrt x}{\sqrt y(\sqrt y-2)}:\frac{\sqrt x}{3(\sqrt y-2)}$$

   $$=\frac{\sqrt x}{\sqrt y(\sqrt y-2)}\cdot\frac{3(\sqrt y-2)}{\sqrt x}=\frac{3}{\sqrt y}$$

4. $$\frac{\sqrt m}{\sqrt m-\sqrt n}:\left(\frac{\sqrt m+\sqrt n}{\sqrt n}+\frac{\sqrt n}{\sqrt m-\sqrt n}\right)$$

   $$=\frac{\sqrt m}{\sqrt m-\sqrt n}:\frac{(\sqrt m+\sqrt n)(\sqrt m-\sqrt n)+\sqrt n\cdot\sqrt n}{\sqrt n(\sqrt m-\sqrt n)}$$

   $$=\frac{\sqrt m}{\sqrt m-\sqrt n}:\frac{m-n+n}{\sqrt n(\sqrt m-\sqrt n)} =\frac{\sqrt m}{\sqrt m-\sqrt n}:\frac{m}{\sqrt n(\sqrt m-\sqrt n)}$$

   $$=\frac{\sqrt m\cdot\sqrt n(\sqrt m-\sqrt n)}{(\sqrt m-\sqrt n)\cdot m}=\frac{\sqrt n}{\sqrt m}=\sqrt{\frac{n}{m}}$$

5. $$\left(\frac{\sqrt x+1}{\sqrt x-1}-\frac{4\sqrt x}{x-1}\right)\cdot\frac{x+\sqrt x}{\sqrt x-1}$$

   $$=\left(\frac{\sqrt x+1}{\sqrt x-1}-\frac{4\sqrt x}{(\sqrt x-1)(\sqrt x+1)}\right)\cdot\frac{\sqrt x(\sqrt x+1)}{\sqrt x-1}$$

   $$=\frac{(\sqrt x+1)^2-4\sqrt x}{(\sqrt x-1)(\sqrt x+1)}\cdot\frac{\sqrt x(\sqrt x+1)}{\sqrt x-1}$$

   $$=\frac{x+2\sqrt x+1-4\sqrt x}{\sqrt x-1}\cdot\frac{\sqrt x}{\sqrt x-1} =\frac{x-2\sqrt x+1}{\sqrt x-1}\cdot\frac{\sqrt x}{\sqrt x-1}$$

   $$=\frac{(\sqrt x-1)^2}{(\sqrt x-1)^2}\cdot\sqrt x=\sqrt x$$

6. $$\frac{a-64}{\sqrt a+3}\cdot\frac{1}{a+8\sqrt a}-\frac{\sqrt a+8}{a-3\sqrt a}$$

   $$=\frac{(\sqrt a-8)(\sqrt a+8)}{(\sqrt a+3)\cdot\sqrt a(\sqrt a+8)}-\frac{\sqrt a+8}{\sqrt a(\sqrt a-3)}$$

   $$=\frac{\sqrt a-8}{\sqrt a(\sqrt a+3)}-\frac{\sqrt a+8}{\sqrt a(\sqrt a-3)}$$

   $$=\frac{(\sqrt a-8)(\sqrt a-3)-(\sqrt a+8)(\sqrt a+3)}{\sqrt a(\sqrt a+3)(\sqrt a-3)}$$

   $$=\frac{a-3\sqrt a-8\sqrt a+24-a-3\sqrt a-8\sqrt a-24}{\sqrt a(\sqrt a+3)(\sqrt a-3)}$$

   $$=\frac{-22\sqrt a}{\sqrt a(a-9)}=-\frac{22}{a-9}=\frac{22}{9-a}$$

Ответ

1) $$\frac{4}{a+\sqrt a}$$;
2) $$-\frac{1}{\sqrt{ab}}$$;
3) $$\frac{3}{\sqrt y}$$;
4) $$\sqrt{\frac{n}{m}}$$;
5) $$\sqrt x$$;
6) $$\frac{22}{9-a}$$.