Let's simplify the expression \((-4x^{-3})^3\).

### Step 1: Apply the power of a power rule
The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). So for the expression \((-4x^{-3})^3\), apply this rule to both the constant \(-4\) and the term involving \(x\).

\[
(-4x^{-3})^3 = (-4)^3 \cdot (x^{-3})^3
\]

### Step 2: Simplify the terms
1. \((-4)^3 = -64\)
2. \((x^{-3})^3 = x^{-9}\) (Using the power rule: \(a^{m \cdot n} = a^{m \cdot n}\)).

Now, putting it all together:

\[
(-4x^{-3})^3 = -64x^{-9}
\]

### Final result:
\[
-64x^{-9}
\]

### Tip:
Remember that \(x^{-n} = \frac{1}{x^n}\), so you could also express the final result as:
\[
\frac{-64}{x^9}
\]

Would you like further clarification or assistance with this? Let me know if you have more questions!

Here are some follow-up questions:
1. How would you simplify other similar expressions, such as \((3x^2)^4\)?
2. What happens when you multiply terms with negative exponents?
3. Can you simplify an expression with both negative and positive exponents, like \(x^3 \cdot x^{-5}\)?
4. How do you deal with fractional exponents, for example, \(x^{\frac{1}{2}}\)?
5. What are the rules for simplifying expressions with both constants and variables?

Learn how to simplify the expression (-4x^-3)^3 step-by-step using the power of a power rule. This problem involves negative exponents, and applying algebraic rules for exponents to find the final result. Suitable for students in Grades 8-10.

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