№ 7.4 Алгебра = № 12.4 Математика
Спростіть вираз:
1. $\left(\frac{x}{7}\ +\ \frac{7}{x}\ +2\right)\cdot\frac{1}{x+7};$
2. $\left(1+\frac{m}{3n}\right):\left(1-\frac{m}{3n}\right);$
3. $\left(\frac{a}{a+2}-3a\right)\cdot\frac{a+2}{a};$
4. $\left(2+\frac{x}{x+1}\right):\frac{9x+6}{5x^2+5x}.$
Розв'язок:
1. $\left(\frac{x}{7}\ +\ \frac{7}{x}\ +2\right)\cdot\frac{1}{x+7}=$
$= \frac{x^2+14x+49}{7x}\cdot\frac{1}{x+7}=$
$= \frac{\left(x+7\right)^2}{7x\left(x+7\right)}=\frac{x+7}{7x};$
2. $\left(1+\frac{m}{3n}\right):\left(1-\frac{m}{3n}\right)=$
$= \frac{3n+m}{3n}:\frac{3n-m}{3n}=$
$= \frac{\left(3n+m\right)\cdot3n}{3n\cdot\left(3n-m\right)}=\frac{3n+m}{3n-m};$
3. $\left(\frac{a}{a+2}-3a\right)\cdot\frac{a+2}{a}=$
$= \frac{a-{3a}^2-6a}{a+2}\cdot\frac{a+2}{a}=$
$= \frac{\left(-3a^2-5a\right)\left(a+2\right)}{\left(a+2\right)a}=$
$= \frac{a\left(-3a-5\right)}{a}\ =-3a-5;$
4. $\left(2+\frac{x}{x+1}\right):\frac{9x+6}{5x^2+5x}=$
$= \frac{2x+2+x}{x+1}:\frac{9x+6}{5x^2+5x}=$
$= \frac{\left(3x+2\right)\cdot5x\left(x+1\right)}{\left(x+1\right)\cdot3\left(3x+2\right)}=\frac{5x}{3}.$