Certainly! Let's solve the equation step by step.
We start with the given equation:
[tex]\[ 3^{2n} \times 3^3 = 3^9 \][/tex]
First, we use the properties of exponents to combine the terms on the left side. According to the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex], we can add the exponents when multiplying with the same base.
Therefore:
[tex]\[ 3^{2n} \times 3^3 = 3^{2n + 3} \][/tex]
So, the equation now looks like:
[tex]\[ 3^{2n + 3} = 3^9 \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 2n + 3 = 9 \][/tex]
Next, we solve for [tex]\(n\)[/tex]. Subtract 3 from both sides of the equation:
[tex]\[ 2n = 9 - 3 \][/tex]
[tex]\[ 2n = 6 \][/tex]
Now, divide both sides by 2 to isolate [tex]\(n\)[/tex]:
[tex]\[ n = \frac{6}{2} \][/tex]
[tex]\[ n = 3 \][/tex]
Therefore, the value of [tex]\(n\)[/tex] is:
[tex]\[ n = 3 \][/tex]
