Let's break down the expression from the image step by step:

The given expression is:

$\left( \frac{8x^{-2}y^0}{4x^{-3}y^{-3}} \right)^3$

Step 1: Simplify the fraction inside the parentheses

First, simplify the coefficients, $8$ and $4$: $\frac{8}{4} = 2$

Now, simplify the exponents of $x$ and $y$:

For $x$, apply the rule $\frac{x^a}{x^b} = x^{a-b}$: $\frac{x^{-2}}{x^{-3}} = x^{-2 - (-3)} = x^{-2 + 3} = x^1 = x$

For $y$, use the same exponent rule: $\frac{y^0}{y^{-3}} = y^{0 - (-3)} = y^{0 + 3} = y^3$

So, the expression inside the parentheses becomes: $2xy^3$

Step 2: Apply the outer exponent

Now apply the exponent 3 to everything inside the parentheses: $\left(2xy^3\right)^3 = 2^3 \cdot x^3 \cdot (y^3)^3 = 8x^3y^9$

Final Answer:

$8x^3y^9$

Would you like a more detailed explanation of any part of this process? Here are some follow-up questions to explore:

1. What are the rules of exponents used in simplifying fractions with variables?
2. How do we apply exponents to expressions with coefficients and variables?
3. How do negative exponents affect simplification?
4. Can you provide more examples of simplifying expressions with exponents?
5. What happens if the outer exponent is negative?

Tip: Always simplify the terms inside parentheses before applying any external exponents!