Question

Find the value of $n.$
a) $5^{10}÷5^8={\left(\frac{1}{5}\right)}^n$
b) ${\left(\frac{-2}{3}\right)}^4÷{\left(\frac{-2}{3}\right)}^3={\left(\frac{-3}{2}\right)}^n$
c) ${\left(-5\right)}^4÷{\left(-5\right)}^2=5^n$

Answer 1

$a)5^{10}÷5^8={\left(\frac{1}{5}\right)}^n$

$\Rightarrow$ $\frac{5^{10}}{5^8}={\left(\frac{1}{5}\right)}^n\left[\because \frac{a^m}{a^n}=a^{m-n}\right]$

$\Rightarrow$ $5^{10-8}={\left(5\right)}^{-n}$

Comparing, $10-8=-n$

$\Rightarrow$ $2=-n$

$n=-2$

$b){\left(\frac{-2}{3}\right)}^4÷{\left(\frac{-2}{3}\right)}^3={\left(\frac{-3}{2}\right)}^n$

$\Rightarrow$ ${\left(\frac{-2}{3}\right)}^4\times {\left(\frac{3}{-2}\right)}^3={\left(\frac{-3}{2}\right)}^n$

$\Rightarrow$ $\frac{+2}{3}\times \frac{+2}{3}\times \frac{+2}{3}\times \frac{+2}{3}\times \frac{3}{-2}\times \frac{3}{-2}\times \frac{3}{-2}={\left(\frac{-3}{2}\right)}^n$

$\Rightarrow \left(\frac{-2}{3}\right)={\left(\frac{-3}{2}\right)}^n$

$\Rightarrow {\left(\frac{-3}{2}\right)}^{-1}={\left(\frac{-3}{2}\right)}^n$

$\therefore n=-1$

$c){\left(-5\right)}^4÷{\left(-5\right)}^2=5^n$

$\Rightarrow$ $\frac{{\left(-5\right)}^4}{{\left(-5\right)}^2}={\left(5\right)}^n$

$\Rightarrow$ ${\left(-5\right)}^{4-2}=5^n\left[\because \frac{a^m}{a^n}=a^{m-n}\right]$

$\Rightarrow$ ${\left(-5\right)}^2={\left(5\right)}^n$

$\Rightarrow$ $-5\times -5=25=5^2=5^n$

$n=2$