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Simplify the following:$ \frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-2^{2} \times 5^{n}} $
Given:
\( \frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-2^{2} \times 5^{n}} \)
To do:
We have to simplify the given expression.
Solution:
We know that,
$(a^{m})^{n}=a^{m n}$
$a^{m} \times a^{n}=a^{m+n}$
$a^{m} \div a^{n}=a^{m-n}$
$a^{0}=1$
$\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-2^{2} \times 5^{n}}=\frac{5^{n}(5^{3}-6 \times 5^{1})}{5^{n}(9-2^{2})}$
$=\frac{125-30}{9-4}$
$=\frac{95}{5}$
$=19$
Therefore, $\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^{n}-2^{2} \times 5^{n}}=19$.
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