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The problem is asking for the least common denominator (LCD) of the given rational expressions. The denominators are:

1. \(6ab^3\)
2. \(2a^2\)
3. \(4b^2\)

### Steps to find the LCD:

1. **Factor each denominator:**

   - \( 6ab^3 = 2 \cdot 3 \cdot a \cdot b^3 \)
   - \( 2a^2 = 2 \cdot a^2 \)
   - \( 4b^2 = 2^2 \cdot b^2 \)

2. **Identify the highest powers of all factors:**

   - For \(2\), the highest power is \(2^2\) (from \(4b^2\)).
   - For \(3\), it appears in \(6ab^3\).
   - For \(a\), the highest power is \(a^2\) (from \(2a^2\)).
   - For \(b\), the highest power is \(b^3\) (from \(6ab^3\)).

3. **Multiply the highest powers of all factors:**

   - \( 2^2 \cdot 3 \cdot a^2 \cdot b^3 = 4 \cdot 3 \cdot a^2 \cdot b^3 = 12a^2b^3 \)

Thus, the **least common denominator (LCD)** is \( \mathbf{12a^2b^3} \).

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Do you want more details or have any questions?

Here are some related questions:
1. How do you add or subtract rational expressions with this LCD?
2. What if one of the expressions has a different variable not seen in the others?
3. How do you factor more complex denominators for LCD?
4. How can this process be applied to more than three rational expressions?
5. What is the process for finding the greatest common factor (GCF) of these denominators?

**Tip:** When finding the LCD, always take the highest power of each factor that appears in any of the denominators.

What is the least common denominator of each list of rational expressions?

Learn how to find the least common denominator (LCD) for a list of rational expressions like 5/(6ab^3), -7/(2a^2), and 9/(4b^2). This guide walks through factoring denominators and determining the highest powers of variables and constants. Ideal for students in Grades 8-10.

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