Question

Find number of triplets $(a,b,c)$ possible such that $(a^b{)}^c=4$.If $a,b,c$ are positive integers only.

Answer 1

$(a^b{)}^c=4$

$a^{bc}=4^1$ (or) $2^2$

$\underset{–}{Case\left(i\right)}$ When $a=4$ then $bc=1$

I.e., $b=c=1$

$\underset{–}{Case\left(ii\right)}$ When $a=2$ then $bc=2$

I.e., $b=1$ and $c=2$ (or) $b=2$ and $c=1$

The possible triplets $(a,b,c)$ are $(4,1,1),(2,1,2),(2,2,1)$

$\therefore$ The number of such triplets = 3