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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 \( 6x^2 y^3 \).

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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.

Find the least common denominator (LCD) of the following rational expressions: (x + y) / (6y^3) and (3x) / (2x^2 y)

Learn how to find the least common denominator (LCD) of rational expressions (x + y) / (6y^3) and (3x) / (2x^2 y) with a detailed step-by-step solution. This algebraic problem demonstrates how to take the highest powers of all terms in the denominators. Suitable for students in Grades 8-10.

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