To solve the equation [tex]\(9^n + 9^n + 9^n = 3^{2013}\)[/tex] and find the value of [tex]\(n\)[/tex], let's go step by step:
1. Combine the terms on the left-hand side:
We have three terms of [tex]\(9^n\)[/tex], so we can combine them as follows:
[tex]\[
9^n + 9^n + 9^n = 3 \cdot 9^n
\][/tex]
2. Express [tex]\(9^n\)[/tex] with a base of 3:
Notice that [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex]. Therefore, [tex]\(9^n\)[/tex] can be written using the base 3:
[tex]\[
9^n = (3^2)^n = 3^{2n}
\][/tex]
Substituting this into our equation gives:
[tex]\[
3 \cdot 9^n = 3 \cdot 3^{2n} = 3^{1 + 2n}
\][/tex]
3. Set the exponents equal to each other:
Now that both sides of the equation have the same base, we can set their exponents equal to each other:
[tex]\[
3^{1 + 2n} = 3^{2013}
\][/tex]
This yields:
[tex]\[
1 + 2n = 2013
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
To find [tex]\(n\)[/tex], solve the linear equation:
[tex]\[
1 + 2n = 2013 \implies 2n = 2013 - 1 \implies 2n = 2012 \implies n = \frac{2012}{2} = 1006
\][/tex]
Thus, the value of [tex]\(n\)[/tex] is [tex]\(\boxed{1006}\)[/tex].
