Question

Solve the equation ${\left(\frac{1}{9}\right)}^m={81}^{m+4}$. Check your solution.

Short answer

The solution for the equation is $m=-\frac{8}{3}$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is ${\left(\frac{1}{9}\right)}^m={81}^{m+4}$.

Step 2. Simplify the equation.

The given equation ${\left(\frac{1}{9}\right)}^m={81}^{m+4}$is solved as:

$\begin{array}{l}{\left(\frac{1}{9}\right)}^m={81}^{m+4}\mathrm{....}\left(\text{Given}\right)\\ {\left(9\right)}^{-m}={\left(9^2\right)}^{m+4}\\ {\left(9\right)}^{-m}={\left(9\right)}^{2m+8}\\ -m=2m+8\\ 3m=-8\\ m=-\frac{8}{3}\end{array}$

Step 3. Verify the result for obtained value.

Plugging $m=-\frac{8}{3}$in given equation and verify the result.

$\begin{array}{l}{\left(\frac{1}{9}\right)}^m={81}^{m+4}\mathrm{....}\left(\text{Given}\right)\\ {\left(\frac{1}{9}\right)}^{\left(-\frac{8}{3}\right)}={81}^{\left(-\frac{8}{3}\right)+4}\\ {\left(9\right)}^{\left(\frac{8}{3}\right)}={\left(9^2\right)}^{\left(\frac{4}{3}\right)}\\ 9^{\frac{8}{3}}=9^{\frac{8}{3}}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $m=-\frac{8}{3}$.

Step 4. Conclusion.

The solution for the given equation is $m=-\frac{8}{3}$.