Question

Find the value of x for which ${\left(\frac{5}{3}\right)}^{-4}\times {\left(\frac{5}{3}\right)}^{-5}={\left(\frac{5}{3}\right)}^{3x}.$

Answer 1

Consider the left side:
${\left(\frac{5}{3}\right)}^{-4}\times {\left(\frac{5}{3}\right)}^{-5}={\left(\frac{5}{3}\right)}^{(-4+(-5\left)\right)}={\left(\frac{5}{3}\right)}^{-9}$

Given:
${\left(\frac{5}{3}\right)}^{-9}={\left(\frac{5}{3}\right)}^{3x}$
Comparing the powers:
$-9=3x\Rightarrow x=-3$