№ 18.37 Алгебра = № 34.37 Математика
Спростіть вираз:
1. $\frac{\sqrt m+1}{m\sqrt m+m+\sqrt m}:\frac{1}{m^2-\sqrt m};$
2. $\frac{a+b}{\sqrt{ab}-b}-\frac{2\sqrt a}{\sqrt a-\sqrt b};$
3. $\left(\frac{\sqrt x}{\sqrt x+\sqrt y}-\frac{\sqrt x-\sqrt y}{\sqrt x}\right):\sqrt{\frac{y}{x}}.$
4. $\frac{a+b}{\sqrt2{ab}+2a}+\frac{\sqrt b}{\sqrt a+\sqrt b}.$
Розв'язок:
1. $\frac{\sqrt m+1}{m\sqrt m+m+\sqrt m}:\frac{1}{m^2-\sqrt m}=$
$= \frac{(\sqrt m+1)(\sqrt m-1)}{(\sqrt m-1)\left(m\sqrt m+(\sqrt m)^2+\sqrt m\right)}\cdot$
$\cdot\frac{m^2-\sqrt m}{1}=$
$= \frac{m-1}{\sqrt m(\sqrt m-1)(m+\sqrt m+1)}\cdot$
$\cdot\frac{m^2-\sqrt m}{1}=$
$= \frac{(m-1)\cdot\left(m^2-\sqrt m\right)}{\sqrt m\left(\sqrt{m^3}-1\right)}=$
$= \frac{(m-1)\left(m^2-\sqrt m\right)}{\sqrt{m^4}-\sqrt m}=$
$= \frac{(m-1)\left(m^2-\sqrt m\right)}{\sqrt{\left(m^2\right)^2}-\sqrt m}=$
$= \frac{(m-1)\left(m^2-\sqrt m\right)}{m^2-\sqrt m}=m-1;$
2. $\frac{a+b}{\sqrt{ab}-b}-\frac{2\sqrt a}{\sqrt a-\sqrt b}=$
$= \frac{a+b}{\sqrt{ab}-(\sqrt b)^2}-\frac{2\sqrt a}{\sqrt a-\sqrt b}=$
$= \frac{a+b}{\sqrt b(\sqrt a-\sqrt b)}-\frac{2\sqrt a}{\sqrt a-\sqrt b}=$
$= \frac{a+b-2\sqrt a\sqrt b}{\sqrt b(\sqrt a-\sqrt b)}=$
$= \frac{(\sqrt a)^2-2\sqrt{ab}+(\sqrt b)^2}{\sqrt b(\sqrt a-\sqrt b)}=$
$= \frac{(\sqrt a-\sqrt b)^2}{\sqrt b(\sqrt a-\sqrt b)}= \frac{\sqrt a-\sqrt b}{\sqrt b};$
3. $\left(\frac{\sqrt x}{\sqrt x+\sqrt y}-\frac{\sqrt x-\sqrt y}{\sqrt x}\right):\sqrt{\frac{y}{x}}=$
$= \frac{\sqrt y}{\sqrt x+\sqrt y};$
а) $\frac{\sqrt x}{\sqrt x+\sqrt y}-\frac{\sqrt x-\sqrt y}{\sqrt x}=$
$= \frac{\left({\sqrt x}^2\right)-(\sqrt x-\sqrt y)(\sqrt x+\sqrt y)}{\sqrt x(\sqrt x+\sqrt y)}=$
$= \frac{x-\left((\sqrt x)^2-(\sqrt y)^2\right)}{\sqrt x(\sqrt x+\sqrt y)}=$
$= \frac{x-(x-y)}{\sqrt x(\sqrt x+\sqrt y)}=$
$= \frac{x-x+y}{\sqrt x(\sqrt x+\sqrt y)}=\frac{y}{\sqrt x(\sqrt x+\sqrt y)};$
б) $\frac{y}{\sqrt x(\sqrt x+\sqrt y)}:\frac{\sqrt y}{x}=$
$= \frac{y\cdot\sqrt x}{\sqrt x\cdot\sqrt y(\sqrt x+\sqrt y)}=$
$= \frac{y}{\sqrt y(\sqrt x+\sqrt y)}=$
$= \frac{y\sqrt y}{\sqrt y\cdot\sqrt y(\sqrt x+\sqrt y)}=$
$= \frac{y\sqrt y}{y(\sqrt x+\sqrt y)}=\frac{\sqrt y}{\sqrt x+\sqrt y}..$
4. $ \ \frac{a+b}{2\sqrt{ab}+2a}+\frac{\sqrt b}{\sqrt a+\sqrt b}=$
$= \frac{a+b}{2\left(\sqrt{ab}+a\right)}+\frac{\sqrt b}{\sqrt a+\sqrt b}=$
$=\frac{a+b}{2\sqrt a\left(\sqrt b+\sqrt a\right)}+\frac{\sqrt b}{\sqrt a+\sqrt b}=$
$= \frac{a+b}{2\sqrt a\left(\sqrt a+\sqrt b\right)}+\frac{2\sqrt{a\sqrt b}}{2\sqrt a\left(\sqrt a+\sqrt b\right)}=$
$=\frac{a+b+2\sqrt{ab}}{2\sqrt a\left(\sqrt a+\sqrt b\right)}=$
$= \frac{\left(\sqrt a+\sqrt b\right)^2}{2\sqrt a\left(\sqrt a+\sqrt b\right)}= \frac{\sqrt a+\sqrt b}{2\sqrt a}.$