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 $3$.   

Tutorialspoint

Updated on: 10-Oct-2022

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