Упр.955 ГДЗ Никольский Потапов 9 класс (Алгебра)

а) (vx/(vx+1)+1):(1-vx/(vx+1)); б) (va-1/(1+va))·(va+1)/(1-va-a);
в) (x-y)/(vx-vy)-(xvx-yvy)/(x-y); г) vm/(vm-7)-4/(vm+7)+m/(49-m).

1. а)

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

   $$=\frac{2\sqrt{x}+1}{\sqrt{x}+1}:\frac{1}{\sqrt{x}+1}=2\sqrt{x}+1.$$

2. б)

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

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

3. в)

   $$\frac{x-y}{\sqrt{x}-\sqrt{y}}-\frac{x\sqrt{x}-y\sqrt{y}}{x-y}= \frac{(x-y)(\sqrt{x}+\sqrt{y})-(x\sqrt{x}-y\sqrt{y})}{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}$$

   $$=\frac{x\sqrt{y}-y\sqrt{x}}{x-y}= \frac{\sqrt{xy}(\sqrt{x}-\sqrt{y})}{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})} =\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}.$$

4. г)

   $$\frac{\sqrt{m}}{\sqrt{m}-7}-\frac{4}{\sqrt{m}+7}+\frac{m}{49-m}= \frac{\sqrt{m}}{\sqrt{m}-7}-\frac{4}{\sqrt{m}+7}+\frac{m}{(7-\sqrt{m})(7+\sqrt{m})}$$

   $$=\frac{\sqrt{m}(\sqrt{m}+7)-4(\sqrt{m}-7)-m}{(\sqrt{m}-7)(\sqrt{m}+7)}$$

   $$=\frac{m+7\sqrt{m}-4\sqrt{m}+28-m}{m-49} =\frac{3\sqrt{m}+28}{m-49}.$$

Ответ

а) $$2\sqrt{x}+1$$; б) $$-1$$; в) $$\frac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}$$; г) $$\frac{3\sqrt{m}+28}{m-49}$$.