# How do you solve 36^(2p)=216^(p-1)?

Nov 16, 2016

$p = - 3$

#### Explanation:

Before solving this exponential equation we can recognize that
$\text{ }$
$\textcolor{red}{36} \text{ and " color(red)216 " }$ can be written as powers of base $\text{ } \textcolor{red}{6}$
$\text{ }$
color(red)(36 = 6^2
$\text{ }$
color(red)(216 = 6^3
$\text{ }$
${36}^{2 p} = {216}^{p - 1}$
$\text{ }$
$\Rightarrow {\textcolor{red}{\left({6}^{2}\right)}}^{2 p} = {\textcolor{red}{\left({6}^{3}\right)}}^{p - 1}$
$\text{ }$
$\Rightarrow {6}^{4 p} = {6}^{3 \left(p - 1\right)}$
$\text{ }$
Therefore,
$\text{ }$
$4 p = 3 \left(p - 1\right)$
$\text{ }$
$\Rightarrow 4 p = 3 p - 3$
$\text{ }$
$\Rightarrow 4 p - 3 p = - 3$
$\text{ }$
$\Rightarrow p = - 3$