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To simplify the expression \(\left(\frac{2x^{-3}}{4x^{-5}}\right)^3\) using exponent rules, we can follow these steps:

1. **Simplify the fraction inside the parentheses**:
   \[
   \frac{2x^{-3}}{4x^{-5}} = \frac{2}{4} \cdot x^{-3-(-5)}
   \]
   \[
   = \frac{1}{2} \cdot x^{2}
   \]

2. **Apply the exponent outside the parentheses**:
   \[
   \left(\frac{1}{2}x^2\right)^3 = \left(\frac{1}{2}\right)^3 \cdot (x^2)^3
   \]
   \[
   = \frac{1}{8} \cdot x^{6}
   \]

The simplified expression is \(\frac{x^6}{8}\).

Do you need further details or have any questions about the steps? Here are a few follow-up topics we can explore:

1. What happens if the exponents inside the parentheses are different?
2. How do we simplify expressions with higher exponents?
3. What are the basic exponent rules used in simplification?
4. How would changing the value of the base (other than 2 and 4) affect the simplification process?
5. Can this method be applied to simplifying expressions with variables other than x?

**Tip:** Remember that when simplifying expressions with negative exponents, it can be helpful to first rewrite them in a form with positive exponents before applying any further operations.

Learn how to simplify the exponent expression (2x^-3 / 4x^-5)^3 step by step using exponent rules. This problem demonstrates key concepts in algebra, including the power of a quotient and properties of exponents. Suitable for students in Grades 9-11.

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