The problem asks to find the values of $a$ and $b$ such that the second expression is equivalent to the first.

The two expressions given are:

[ \sqrt[3]{\frac{128xy^5}{32x}} \quad \text{and} \quad \sqrt[3]{\frac{63y^5}{ax^b}} ]

Step-by-step Solution:

1. Simplify the first expression: $\frac{128xy^5}{32x} = \frac{128}{32} \cdot \frac{xy^5}{x} = 4y^5$ So the first expression becomes: $\sqrt[3]{4y^5}$

2. The second expression is already given as: $\sqrt[3]{\frac{63y^5}{ax^b}}$

3. For the expressions to be equivalent, we compare the simplified first expression with the second: [ \sqrt[3]{4y^5} = \sqrt[3]{\frac{63y^5}{ax^b}} ]

4. The $y^5$ terms are already identical on both sides, so the remaining terms must be equal: $4 = \frac{63}{ax^b}$

5. From this, solve for $a$ and $b$: $ax^b = \frac{63}{4} = 15.75$

   Since there is no $x$ term on the left-hand side (the expression for $4y^5$), we know $b = 0$.

6. Finally, solve for $a$: $a = 15.75$

Thus, the values of $a$ and $b$ are: $a = 15.75 \quad \text{and} \quad b = 0$

If you have any questions or would like more details, feel free to ask!

Here are 5 related questions you can try:

1. What happens if the variable $x$ were not simplified in the first expression?
2. How would the solution change if the exponents on $y$ were different?
3. Can you generalize this method for other similar expressions involving roots?
4. How can the cube root be manipulated if different numbers were used in the numerator or denominator?
5. What role does the assumption $x > 0$ and $y \approx 0$ play in the solution?

Tip: Always check if variables can be simplified before equating two expressions—it can save a lot of calculation!