Question

If $\frac{5^m\times 5^3\times 5^{-2}}{5^{-5}}=5^{12},$ then find m.

Answer 1

We know that, $a^m\times a^n=a^{m+n}and{a}^{-m}=\frac{1}{a^m}$

$Given,\frac{5^m\times 5^3\times 5^{-2}}{5^{-5}}=5^{12}$
$\Rightarrow 5^m\times 5^3\times 5^{-2}\times 5^5=5^{12}\Rightarrow 5^m\times 5^{[3+(-2)+5]}=5^{12}\Rightarrow 5^m\times 5^6=5^{12}\Rightarrow 5^{m+6}=5^{12}[\because a^m\times a^n=a^{m+n}]$

Since the base are same on both sides we can equate the exponents.
$\Rightarrow m+6=12$
$\Rightarrow m=6$