We are given the expression:

\[
\frac{(n^{-3})^4}{n^{-15}}
\]

To simplify this, let's break it down step by step:

### Step 1: Apply the power of a power rule
The rule \((a^m)^n = a^{m \cdot n}\) allows us to simplify \((n^{-3})^4\). So:

\[
(n^{-3})^4 = n^{-12}
\]

### Step 2: Substitute this back into the original expression
Now we have:

\[
\frac{n^{-12}}{n^{-15}}
\]

### Step 3: Use the quotient of powers rule
The rule \(\frac{a^m}{a^n} = a^{m-n}\) can be applied here. So:

\[
n^{-12 - (-15)} = n^{-12 + 15} = n^3
\]

Thus, the simplified expression is:

\[
n^3
\]

### Final Answer:

\[
n^3
\]

Would you like further clarification or more details on this solution?

Here are 5 related questions to further your understanding:

1. How would you simplify \((n^2)^5 / n^3\)?
2. What happens if the powers are positive, e.g., \((n^3)^2 / n^5\)?
3. How do you apply these rules to more complex expressions like \(\frac{(2n^{-2})^3}{n^{-6}}\)?
4. Can you explain the difference between \(a^m \cdot a^n\) and \(\frac{a^m}{a^n}\)?
5. How would you simplify a more general expression \(\frac{(x^{-a})^b}{x^{-c}}\)?

### Tip:
Always remember the power of a power rule and the quotient of powers rule to quickly simplify such expressions!

This detailed solution walks through simplifying the expression (n^-3)^4 / n^-15 using exponent rules such as the power of a power and quotient of powers. Learn how to apply these rules step-by-step for a clear and concise solution. Suitable for Grades 9-11 students.

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