№ 4.27 Алгебра = № 4.27 Математика
Спростіть вираз:
1) $\ \frac{a+4}{ab-a^2}+\frac{b+4}{ab-{\ b}^2}; $
2) $\frac{m^2}{mx-x^2}+\frac{x}{x-m};$
3) $\ \frac{2}{x^2-4}-\frac{1}{x^2+2x};$
4)$\ \frac{3ab-27a^2}{b^2-3ab}-\frac{3a^2-b^2}{ab-3a^2}.$
Розв'язок:
1) $\ \frac{a+4}{ab-a^2}+\frac{b+4}{ab-{\ b}^2}=$
$= \frac{a+4}{a\left(b-a\right)}-\frac{b+4}{b\left(b-a\right)}=$
$= \frac{ab+4b-ab+4a}{ab\left(b-a\right)}=$
$= \frac{4\left(b-a\right)}{ab\left(b-a\right)} = \frac{4}{ab}; $
2) $\frac{m^2}{mx-x^2}+\frac{x}{x-m} =$
$\frac{m^2}{x\left(m-x\right)}-\frac{x}{m-x}= \frac{m^2-x^2}{x\left(m-x\right)}=$
$= \frac{\left(m-x\right)\left(m+x\right)}{x\left(m-x\right)}=\frac{m+x}{x};$
3)$\frac{2}{x^2-4}-\frac{1}{x^2+2x}=$
$= \frac{2}{\left(x-2\right)\left(x+2\right)}-\frac{1}{x\left(x+2\right)}=$
$= \frac{2x-\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}=\frac{2x-x+2}{x\left(x-2\right)\left(x+2\right)}=$
$= \frac{x+2}{x\left(x-2\right)\left(x+2\right)}=\frac{1}{x^2-2x};$
4)$\ \frac{3ab-27a^2}{b^2-3ab}-\frac{3a^2-b^2}{ab-3a^2}=$
$= \frac{3ab-27a^2}{b\left(b-3a\right)}-\frac{3a^2-\ b^2}{a\left(b-3a\right)}=$
$= \frac{3a^2b-27a^3-3a^2b+b^3}{ab\left(b-3a\right)}=$
$= \frac{b^3-27a^3}{ab\left(b-3a\right)}= $
$= \frac{\left(b-3a\right)\left(b^2+3ab+9a^2\right)}{ab\left(b-3a\right)}=$
$= \frac{b^2+3ab+9a^2}{ab}.$