Упр.48 Повторение ГДЗ Колмогоров 10-11 класс (Алгебра)

а) $$\left(\frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2a}+2}+\frac{2}{a-\sqrt{2a}}\right)\cdot\frac{\sqrt a-\sqrt2}{a+2}$$

$$\left(\frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt2(\sqrt a+\sqrt2)}+\frac{2}{\sqrt a(\sqrt a-\sqrt2)}\right)\cdot\frac{\sqrt a-\sqrt2}{a+2}$$

$$=\frac{a^2-4-a^2+a\sqrt{2a}+4+2\sqrt{2a}}{\sqrt{2a}(\sqrt a+\sqrt2)(\sqrt a-\sqrt2)}\cdot\frac{\sqrt a-\sqrt2}{a+2}$$

$$=\frac{\sqrt{2a}(a+2)}{\sqrt{2a}(\sqrt a+\sqrt2)(a+2)}=\frac1{\sqrt a+\sqrt2}$$

б) $$\left(\frac{a\sqrt a+b\sqrt b}{\sqrt a+\sqrt b}-\sqrt{ab}\right)\left(\frac{\sqrt a+\sqrt b}{a-b}\right)^2$$

$$=\frac{a\sqrt a+b\sqrt b-a\sqrt b-b\sqrt a}{\sqrt a+\sqrt b}\cdot\left(\frac{\sqrt a+\sqrt b}{a-b}\right)^2$$

$$=\frac{a(\sqrt a-\sqrt b)-b(\sqrt a-\sqrt b)}{\sqrt a+\sqrt b}\cdot\left(\frac{\sqrt a+\sqrt b}{a-b}\right)^2$$

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

в) $$\left(\frac1{\sqrt p+\sqrt{p+1}}+\frac1{\sqrt p-\sqrt{p-1}}\right):\left(1+\sqrt{\frac{p+1}{p-1}}\right)$$

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

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

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

г) $$\left(\frac{\sqrt c}{2}-\frac1{2\sqrt c}\right)^2\left(\frac{\sqrt c-1}{\sqrt c+1}-\frac{\sqrt c+1}{\sqrt c-1}\right)$$

$$=\left(\frac{c-1}{2\sqrt c}\right)^2\cdot\frac{c-2\sqrt c+1-c-2\sqrt c-1}{(\sqrt c+1)(\sqrt c-1)}$$

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

Ответ

а) $$\frac1{\sqrt a+\sqrt2}$$; б) $$1$$; в) $$\sqrt{p-1}$$; г) $$\frac{1-c}{\sqrt c}$$.