Question

If $2^{n-7}\times 5^{n-4}=1250$ find the value of $n$.

Answer 1

Given $2^{n-7}\times 5^{n-4}=1250$
$\Rightarrow 2^{n-7}\times 5^{n-4}=2\times 5^4$ [Since,$1250=2\times 5^4$]
Multplying $5^{-3}$ on both sides, we get
$\Rightarrow 2^{n-7}\times 5^{n-4}\times 5^{-3}=2\times 5^4\times 5^{-3}$
$\Rightarrow 2^{n-7}\times 5^{n-4-3}=2\times 5^{4-3}$
$\Rightarrow 2^{n-7}\times 5^{n-7}=2\times 5$
$\Rightarrow (2\times 5{)}^{n-7}=(10{)}^1$
$\Rightarrow (10{)}^{n-7}={10}^1$
Since, the bases are equal, so we can equate their power as
$\Rightarrow n-7=1$
$\Rightarrow n=1+7$
$\therefore n=8$
Hence, the value of $n=8$