Let's go through the expression carefully to verify the solution.

The expression provided is:

$\left( \frac{y^{1/3} y^{-2}}{5(x^3 y^3)^{3/2}} \right)^{3/2}$

To solve this step-by-step, let’s break down each component and simplify.

1. Simplify the numerator:

   • The numerator is $y^{1/3} y^{-2}$.
   • Combine the exponents for $y$: $y^{1/3 - 2} = y^{1/3 - 6/3} = y^{-5/3}$.

   So, the numerator becomes $y^{-5/3}$.

2. Simplify the denominator:

   • The denominator is $5(x^3 y^3)^{3/2}$.
   • Apply the $3/2$ power to both $x^3$ and $y^3$:
     • $(x^3)^{3/2} = x^{9/2}$,
     • $(y^3)^{3/2} = y^{9/2}$.

   Therefore, the denominator becomes $5 \cdot x^{9/2} y^{9/2}$.

3. Combine numerator and denominator:

   • We now have: $\frac{y^{-5/3}}{5 x^{9/2} y^{9/2}}.$

   • Simplify the $y$-terms by subtracting exponents: $y^{-5/3 - 9/2}$.

   • Find a common denominator for $-5/3$ and $-9/2$:

     • $-5/3 = -10/6$,
     • $-9/2 = -27/6$.

   So, $y^{-5/3 - 9/2} = y^{-(10/6 + 27/6)} = y^{-37/6}$.

   This simplifies to: $\frac{y^{-37/6}}{5 x^{9/2}}.$

4. Apply the outer $3/2$ exponent:

   • Now we raise each term in the fraction to the $3/2$ power: $\left( \frac{y^{-37/6}}{5 x^{9/2}} \right)^{3/2}.$
   • This becomes: $\frac{y^{-37/6 \cdot 3/2}}{5^{3/2} x^{9/2 \cdot 3/2}}.$
   • Simplify each part:
     • $y^{-37/6 \cdot 3/2} = y^{-111/12} = y^{-37/4}$,
     • $x^{9/2 \cdot 3/2} = x^{27/4}$,
     • $5^{3/2}$ remains as it is (which is $\sqrt{125}$ or approximately $11.18$).

   Therefore, the simplified expression is: $\frac{y^{-37/4}}{5^{3/2} x^{27/4}}.$

So, the correct simplified answer is:

$\frac{y^{-37/4}}{5^{3/2} x^{27/4}}.$

Would you like further clarification on any of the steps, or do you have additional questions?

Here are some related questions you might consider:

1. How do we simplify expressions with fractional exponents?
2. What are the rules for handling negative exponents?
3. How can we convert between radical and exponent forms?
4. Why do we apply the exponent to each component in the fraction?
5. What is the significance of common denominators when simplifying exponents?

Tip: Always double-check each step in a complex simplification to avoid errors, especially when combining fractional exponents.