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Solve the following equations:
\( 3^{x-1} \times 5^{2 y-3}=225 \)
Given:
\( 3^{x-1} \times 5^{2 y-3}=225 \)
To do:
We have to solve the given equation.
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,
$3^{x-1} \times 5^{2 y-3}=225$
$\Rightarrow 3^{x-1} \times 5^{2 y-3}=(15)^{2}$
$\Rightarrow 3^{x-1} \times 5^{2 y-3}=(3 \times 5)^{2}$
$\Rightarrow 3^{x-1} \times 5^{2 y-3}=3^{2} \times 5^{2}$
Comparing both sides, we get,
$x-1=2$
$\Rightarrow x=2+1=3$
$2 y-3=2$
$\Rightarrow 2 y=2+3=5$
$\Rightarrow y=\frac{5}{2}$
The values of $x$ and $y$ are $3$ and $\frac{5}{2}$ respectively.
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