We are asked to find the value of $n$ in the equation $(9^n)^4 = 9^{12}$.

1. Problem Description

We are asked to find the value of $n$ in the equation $(9^n)^4 = 9^{12}$.

2. Solution Steps

We use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. Applying this rule to the left side of the equation, we have:

$(9^n)^4 = 9^{n \cdot 4} = 9^{4n}$

So the equation becomes:

$9^{4n} = 9^{12}$

Since the bases are equal, we can equate the exponents:

$4n = 12$

Now, we solve for $n$ by dividing both sides by 4:

$\frac{4n}{4} = \frac{12}{4}$

$n = 3$

3. Final Answer

The value of $n$ is 3.