Question:
Find the value of $x$ for which $\left(\frac{5}{3}\right)^{-4} \times\left(\frac{5}{3}\right)^{-5}=\left(\frac{5}{3}\right)^{3 x}$.
Solution:
Consider the left side:
$\left(\frac{5}{3}\right)^{-4} \times\left(\frac{5}{3}\right)^{-5}=\left(\frac{5}{3}\right)^{(-4+(-5))}=\left(\frac{5}{3}\right)^{-9}$
Given:
$\left(\frac{5}{3}\right)^{-9}=\left(\frac{5}{3}\right)^{3 x}$
Comparing the powers:
$-9=3 x \Rightarrow x=-3$