№ 10.32 Алгебра = № 20.32 Математика
Скоротіть дріб:
1. $\frac{18}{4^{n+1}-4^n};$
2. $\frac{x^9+x^5}{x^{-3}+x};$
3. $\frac{b^{-5}+3-b^2}{3b^3-b^5+b^{-2}}.$
Розв'язок:
1. $\frac{18}{4^{n+1}-4^n}=\frac{18}{4\cdot4^n-4^n}=$
$= \frac{18}{4^n(4-1)}=\frac{18}{4^n\cdot3}=\frac{6}{4^n};$
2. $\frac{x^9+x^5}{x^{-3}+x}=\frac{x^5\left(x^4+1\right)}{\frac{1}{x^3}+x}=$
$= \frac{x^5\left(x^4+1\right)}{\frac{1+x^4}{x^3}}=\frac{x^3\cdot x^5\left(1+x^4\right)}{1+x^4}=$
$= x^3\cdot x^5=x^8;$
3. $\frac{b^{-5}+3-b^2}{3b^3-b^5+b^{-2}}=\frac{\frac{1}{b^5}+3-b^2}{3b^3-b^5+\frac{1}{b^2}}=$
$= \frac{1+3b^5-b^7}{b^5}:\frac{3b^5-b^7+1}{b^2}=$
$= \frac{b^2\cdot\left(1+3b^5-b^7\right)}{b^5\cdot\left(1+3b^5-b^7\right)}=\frac{1}{b^3}.$