https://file.aiquickdraw.com/mathbot/file/1726243096658wxetgwhm.jpg

The expression shown in the image is:

\[
\left( \frac{4x^{-2}}{5y^{-1}} \right)^{-3}
\]

We are tasked with simplifying this expression and expressing all exponents as positive.

### Step-by-step simplification:

1. **Apply the exponent to both the numerator and the denominator:**

   Using the property \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\), we get:

   \[
   \left( \frac{4x^{-2}}{5y^{-1}} \right)^{-3} = \frac{(4x^{-2})^{-3}}{(5y^{-1})^{-3}}
   \]

2. **Distribute the exponents inside the parentheses:**

   Apply the exponent of \(-3\) to each part of the numerator and the denominator. Using the property \((a^m)^n = a^{m \cdot n}\), we get:

   \[
   \frac{(4^{-3})(x^{-2 \cdot (-3)})}{(5^{-3})(y^{-1 \cdot (-3)})}
   \]

   Simplifying the exponents:

   - \(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\)
   - \(x^{-2 \cdot (-3)} = x^6\)
   - \(5^{-3} = \frac{1}{5^3} = \frac{1}{125}\)
   - \(y^{-1 \cdot (-3)} = y^3\)

   So, the expression becomes:

   \[
   \frac{\frac{1}{64}x^6}{\frac{1}{125}y^3}
   \]

3. **Simplify the fraction:**

   Dividing by a fraction is equivalent to multiplying by its reciprocal:

   \[
   \frac{\frac{1}{64}x^6}{\frac{1}{125}y^3} = \frac{1}{64}x^6 \cdot \frac{125}{1y^3} = \frac{125x^6}{64y^3}
   \]

### Final answer:
\[
\frac{125x^6}{64y^3}
\]

All exponents are now positive, and the expression is fully simplified.

---

Would you like more details or further clarification?

Here are five questions related to this concept:
1. What are the rules for dealing with negative exponents?
2. How does the property \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\) apply when simplifying expressions?
3. What happens when we raise a fraction to a negative exponent?
4. How can we simplify expressions with multiple variables and exponents?
5. How do you convert a negative exponent into a positive one?

**Tip:** When simplifying exponents, always apply the exponent rule to each term carefully, whether it's in the numerator or denominator.

Simplify the given expression: (4x^-2 / 5y^-1)^-3. Express the answer so that all exponents are positive.

Learn how to simplify the algebraic expression (4x^-2 / 5y^-1)^-3. This solution covers key exponent rules and algebraic concepts for students in Grades 9-11.

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