Question

Solve the equation ${36}^{2p}={216}^{p-1}$. Check your solution.

Short answer

The solution for the equation is $p=-3$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is ${36}^{2p}={216}^{p-1}$.

Step 2. Simplify the equation.

The given equation ${36}^{2p}={216}^{p-1}$is solved as:

$\begin{array}{l}{36}^{2p}={216}^{p-1}\mathrm{....}\left(\text{Given}\right)\\ {\left(6^2\right)}^{2p}={\left(6^3\right)}^{p-1}\\ 6^{4p}={\left(6\right)}^{3p-3}\\ 4p=3p-3\\ p=-3\end{array}$

Step 3. Verify the result for obtained value.

Plugging $p=-3$in given equation and verify the result.

$\begin{array}{l}{36}^{2p}={216}^{p-1}\mathrm{....}\left(\text{Given}\right)\\ {36}^{2\left(-3\right)}={216}^{\left(-3\right)-1}\\ {36}^{-6}={216}^{-4}\\ {\left(6^2\right)}^{-6}={\left(6^3\right)}^{-4}\\ {\left(6\right)}^{-12}={\left(6\right)}^{-12}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $p=-3$.

Step 4. Conclusion.

The solution for the given equation is $p=-3$.