Simplify the following:$ \frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-(25)^{n+1}} $

Given:

\( \frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-(25)^{n+1}} \)

To do:

We have to simplify the given expression.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$  

$\frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-(25)^{n+1}}=\frac{5 \times(5^{2})^{n+1}-5^{2} \times 5^{2 n}}{5 \times 5^{2 n+3}-(5^{2})^{n+1}}$

$=\frac{5 \times 5^{2 n} \times 5^{2}-5^{2} \times 5^{2 n}}{5 \times 5^{2 n} \times 5^{3}-5^{2 n} \times 5^{2}}$

$=\frac{5^{2 n}(5 \times 5^{2}-5^{2})}{5^{2 n}(5 \times 5^{3}-5^{2})}$

$=\frac{5^{3}-5^{2}}{5^{4}-5^{2}}$

$=\frac{125-25}{625-25}$

$=\frac{100}{600}$

$=\frac{1}{6}$

Therefore, $\frac{5 \times 25^{n+1}-25 \times 5^{2 n}}{5 \times 5^{2 n+3}-(25)^{n+1}}=\frac{1}{6}$.

Tutorialspoint

Updated on: 10-Oct-2022

41 Views

Kickstart Your Career

Get certified by completing the course

Get Started