№ 7.5 Алгебра = № 12.5 Математика
Спростіть вираз:
1. $\left(\frac{m}{5}+\frac{5}{m}-2\right)\cdot\frac{1}{m-5};$
2. $\left(1-\frac{x}{y}\right):\left(1+\frac{x}{y}\right);$
3. $\left(\frac{b}{b-3}-2b\right)\cdot\frac{b-3}{b};$
4. $\left(3-\frac{m}{m+2}\right):\frac{4m+12}{m^2+2m}.$
Розв'язок:
1. $\left(\frac{m}{5}+\frac{5}{m}-2\right)\cdot\frac{1}{m-5}=$
$= \frac{m^2-10m+25}{5m}\cdot\frac{1}{m-5}=$
$= \frac{\left(m-5\right)^2\cdot1}{5m\cdot\left(m-5\right)}=\frac{m-5}{5m};$
2. $\left(1-\frac{x}{y}\right):\left(1+\frac{x}{y}\right)=$
$= \frac{y-x}{y}:\frac{y+x}{y}=$
$= \frac{\left(y-x\right)\cdot y}{y\cdot\left(y+x\right)}=\frac{y-x}{y+x};$
3. $\left(\frac{b}{b-3}-2b\right)\cdot\frac{b-3}{b}=$
$= \frac{b-2b^2+6b}{b-3}\cdot\frac{b-3}{b}=$
$= \frac{\left(7b-2b^2\right)\left(b-3\right)}{\left(b-3\right)b}=$
$= \frac{b\left(7-2b\right)}{b}=7-2b;$
4. $\left(3-\frac{m}{m+2}\right):\frac{4m+12}{m^2+2m}=$
$= \frac{3m+6-m}{m+2}\cdot\frac{m^2+2m}{4m+12}=$
$= \frac{2m+6}{m+2}\cdot\frac{m^2+2m}{4m+12}=$
$= \frac{2\left(m+3\right)\cdot m\left(m+2\right)}{\left(m+2\right)\cdot4\left(m+3\right)}=\frac{m}{2}.$