Question

If $3^9+3^{12}+3^{15}+3^n$ is a perfect cube (of an integer) where $n\in N,$ then find the value of $n.$

Answer 1

consider, $3^9+3^{12}+3^{15}+3^n$

$\Rightarrow 3^9(1+3^3+3^6+3^{n-9})$

$\Rightarrow 3^9$ is perfect cube & for n

$\Rightarrow (1+3^3+3^6+3^{14-9})$

$=\left(1000\right)$ this is a perfect cube

Hence $n=14$