Question

Solve the equation $2^{n+4}=\frac{1}{32}$. Check your solution.

Short answer

The solution for the equation is $n=-9$.

Step-by-step solution

Step 1. Write down the given expression.

The given equation is $2^{n+4}=\frac{1}{32}$.

Step 2. Simplify the equation.

The given equation $2^{n+4}=\frac{1}{32}$is solved as:

$\begin{array}{l}{2}^{n+4}=\frac{1}{32}\mathrm{....}\left(\text{Given}\right)\\ 2^{n+4}=\frac{1}{{\left(2\right)}^5}\\ {\left(2\right)}^{n+4}={\left(2\right)}^{-5}\\ n+4=-5\\ n=-9\end{array}$

Step 3. Verify the result for obtained value.

Plugging $n=-9$in given equation and verify the result.

$\begin{array}{l}{2}^{n+4}=\frac{1}{32}\mathrm{....}\left(\text{Given}\right)\\ 2^{\left(-9\right)+4}=\frac{1}{32}\mathrm{....}\left(\text{Plugging}n=-9\right)\\ {\left(2\right)}^{-5}=\frac{1}{32}\\ \frac{1}{2^5}=\frac{1}{32}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $n=-9$.

Step 4. Conclusion.

The solution for the given equation is $n=-9$.