To solve the problem, we need to find the least common denominator (LCD) of the two rational expressions given:

$\frac{x + y}{6y^3} \quad \text{and} \quad \frac{3x}{2x^2 y}$

Step-by-Step Solution:

1. Denominators:

   • First expression: $6y^3$
   • Second expression: $2x^2 y$

2. Prime Factorization of Denominators:

   • For $6y^3$:
     $6 = 2 \times 3$, so the prime factorization is $2 \times 3 \times y^3$.
   • For $2x^2 y$:
     The prime factorization is $2 \times x^2 \times y$.

3. Finding the Least Common Denominator (LCD): To find the LCD, we take the highest powers of all the variables and constants in the denominators:

   • For the constant $2$, we take $2$ (highest power is 1).
   • For the constant $3$, we take $3$ (since it's only in the first denominator).
   • For $x$, we take $x^2$ (highest power is $x^2$ from the second denominator).
   • For $y$, we take $y^3$ (highest power is $y^3$ from the first denominator).

   Thus, the LCD is: $\text{LCD} = 6x^2 y^3$

Conclusion:

The least common denominator (LCD) of the given rational expressions is $6x^2 y^3$.

Would you like a more detailed explanation of any step?

Here are five related questions to expand on this topic:

1. How do you simplify rational expressions after finding the LCD?
2. What happens if one of the denominators includes a negative exponent?
3. Can you find the least common denominator for three or more rational expressions?
4. How does the presence of a variable in the numerator affect finding the LCD?
5. What is the general method for simplifying complex fractions?

Tip: Always factor the denominators fully when looking for the least common denominator (LCD) to avoid missing any terms.