To simplify the expression $\frac{\sqrt{9y^3}}{\sqrt[3]{8y^2}}$, we can approach it step by step:

1. Simplify the square root in the numerator: $\sqrt{9y^3} = \sqrt{9} \cdot \sqrt{y^3} = 3 \cdot y^{3/2}$ So, the numerator becomes $3y^{3/2}$.

2. Simplify the cube root in the denominator: [ \sqrt[3]{8y^2} = \sqrt[3]{8} \cdot \sqrt[3]{y^2} = 2 \cdot y^{2/3} ] So, the denominator becomes $2y^{2/3}$.

3. Combine the simplified parts: $\frac{3y^{3/2}}{2y^{2/3}}$ To simplify further, subtract the exponents of $y$ in the numerator and denominator: $y^{\frac{3}{2} - \frac{2}{3}} = y^{\frac{9}{6} - \frac{4}{6}} = y^{\frac{5}{6}}$

   Therefore, the simplified expression is: $\frac{3}{2} y^{5/6}$

So, the final simplified expression is $\frac{3}{2} y^{5/6}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you convert a radical expression into an exponent?
2. What are the rules for simplifying expressions with exponents?
3. How do you add and subtract fractions with different denominators?
4. What is the general process for simplifying a complex fraction?
5. How does the square root of a product work compared to the cube root?

Tip: When dealing with roots and exponents, always try to express the roots as fractional exponents to simplify operations like multiplication and division.