If $ 27^{x}=\frac{9}{3^{x}} $, find $ x $.

Given:

\( 27^{x}=\frac{9}{3^{x}} \)

To do: 

We have to find \( x \).

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,

$27^{x}=\frac{9}{3^{x}}$

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

$\Rightarrow 3^{3 x} \times 3^{x}=3^{2}$

$\Rightarrow 3^{3 x+x}=3^{2}$

$\Rightarrow 3^{4 x}=3^{2}$

Comparing both sides, we get,

$4 x=2$

$x=\frac{2}{4}$

$x=\frac{1}{2}$

The value of $x$ is $\frac{1}{2}$.

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Updated on: 10-Oct-2022

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