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?