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Find the values of $x$ in each of the following:
\( 5^{x-2} \times 3^{2 x-3}=135 \)
Given:
\( 5^{x-2} \times 3^{2 x-3}=135 \)
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,
$5^{x-2} \times 3^{2 x-3}=135$
$\Rightarrow 5^{x} \times 5^{-2} \times 3^{2 x} \times 3^{-3}=135$
$\Rightarrow \frac{5^{x} \times 3^{2 x}}{5^{2} \times 3^{3}}=135$
$\Rightarrow 5^{x}\times3^{2 x}=135 \times 5^{2} \times 3^{3}$
$\Rightarrow 5^{x} \times 3^{x} \times 3^{x}=135 \times 25 \times 27$
$\Rightarrow (5 \times 3 \times 3)^{x}=3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 3 \times 3 \times 3$
$\Rightarrow (45)^{x}=(3 \times 5 \times 3)^{3}$
$\Rightarrow (45)^{x}=(45)^{3}$
Comparing both sides, we get,
$x=3$
The value of $x$ is $3$.