https://file.aiquickdraw.com/mathbot/file/1727049740112w6uqjlkw.jpg

The expression in the image is:

\[
\left(\frac{2x^{-3}}{4x^{-5}}\right)^3
\]

Let's break it down step by step:

### Step 1: Simplify the fraction inside the parentheses.
\[
\frac{2x^{-3}}{4x^{-5}} = \frac{2}{4} \cdot \frac{x^{-3}}{x^{-5}}
\]
This simplifies to:
\[
\frac{1}{2} \cdot x^{(-3) - (-5)} = \frac{1}{2} \cdot x^{2}
\]
So, the fraction simplifies to:
\[
\frac{x^2}{2}
\]

### Step 2: Raise the simplified expression to the power of 3.
\[
\left(\frac{x^2}{2}\right)^3 = \frac{(x^2)^3}{2^3} = \frac{x^6}{8}
\]

### Final answer:
The simplified expression is:
\[
\frac{x^6}{8}
\]

Would you like further clarification or additional details?

Here are 5 related questions to explore:
1. How would the expression change if it were raised to a different power, say 4?
2. What happens if the exponents of \(x\) were different in the original expression?
3. How would you simplify a similar expression with negative exponents in the numerator and denominator?
4. What if there were additional terms in the numerator and denominator of the fraction?
5. How would you handle exponents that are fractions (rational exponents)?

**Tip**: When simplifying exponents, remember that subtracting exponents occurs when dividing powers with the same base.

Simplify the expression: (2x^(-3) / 4x^(-5))^3

Learn how to simplify the expression (2x^(-3) / 4x^(-5))^3 using exponent rules. This step-by-step solution covers key algebraic concepts such as simplifying negative exponents and raising a fraction to a power. Suitable for high school students.

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