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.