Let's solve the equation step-by-step.
We start with the equation:
[tex]\[
4 \cdot (3)^n = 324
\][/tex]
First, we want to isolate the term with the exponent, [tex]\((3)^n\)[/tex]. To do this, we'll divide both sides of the equation by 4:
[tex]\[
\frac{4 \cdot (3)^n}{4} = \frac{324}{4}
\][/tex]
Simplifying both sides, we get:
[tex]\[
(3)^n = 81
\][/tex]
Next, we need to express 81 as a power of the base 3. Observing the numbers, we notice that:
[tex]\[
81 = 3 \times 3 \times 3 \times 3 = 3^4
\][/tex]
So, we can rewrite the equation as:
[tex]\[
(3)^n = 3^4
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
n = 4
\][/tex]
Thus, the solution to the equation [tex]\(4 \cdot (3)^n = 324\)[/tex] is:
[tex]\[
n = 4
\][/tex]
