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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 $3$.
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