Simplify the expression $\frac{n^4}{n^{-3} \cdot (n^4)^{-2}}$.

1. Problem Description

Simplify the expression $\frac{n^4}{n^{-3} \cdot (n^4)^{-2}}$.

2. Solution Steps

First, simplify the denominator:

$(n^4)^{-2} = n^{4 \cdot (-2)} = n^{-8}$

$n^{-3} \cdot (n^4)^{-2} = n^{-3} \cdot n^{-8}$

When multiplying exponential terms with the same base, we add the exponents:

$n^{-3} \cdot n^{-8} = n^{-3 + (-8)} = n^{-11}$

Now, we have $\frac{n^4}{n^{-11}}$.

When dividing exponential terms with the same base, we subtract the exponents:

$\frac{n^4}{n^{-11}} = n^{4 - (-11)} = n^{4 + 11} = n^{15}$

3. Final Answer

$n^{15}$