Find the values of $x$ in each of the following:$ \left(\frac{3}{5}\right)^{x}\left(\frac{5}{3}\right)^{2 x}=\frac{125}{27} $

Given:

\( \left(\frac{3}{5}\right)^{x}\left(\frac{5}{3}\right)^{2 x}=\frac{125}{27} \)

To do: 

We have to find the value of $x$.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

$(\frac{3}{5})^{x}(\frac{5}{3})^{2 x}=\frac{125}{27}$

$\Rightarrow (\frac{5}{3})^{-x} \times (\frac{5}{3})^{2 x}=\frac{5^{3}}{3^{3}}$

$\Rightarrow (\frac{5}{3})^{-x+2 x}=(\frac{5}{3})^{3}$

Comparing both sides, we get,

$-x+2 x=3$

$x=3$

The value of $x$ is $3$.