Question

Find all positive integers $m,n$ such that ${\left(3^m\right)}^n=3^m\times 3^n$

Answer 1

Suppose, $(3^m{)}^n=3^m\times 3^n$

$⟹(3{)}^{mn}=(3{)}^{m+n}$

$⟹mn=m+n$

$⟹mn-m-n=0$

$⟹mn-m-n+1-1=0$

$⟹m(n-1)-(n-1)=1$

$⟹(m-1)(n-1)=1$

$⟹m-1=1$ and $n-1=1$

$⟹m=2$ and $n=2$