To find the Least Common Multiple (LCM) of the terms \( a^2 \times x^2 \), \( 6axy^2 \), and \( 9ay^2 \), we can follow these steps:

### Step 1: Factor each term completely
1. \( a^2 \times x^2 = a^2 x^2 \)
2. \( 6axy^2 = 2 \cdot 3 \cdot a \cdot x \cdot y^2 \)
3. \( 9ay^2 = 3^2 \cdot a \cdot y^2 \)

### Step 2: Identify the factors and their highest powers
- **\( a \)**: The highest power is \( a^2 \) (from \( a^2 x^2 \)).
- **\( x \)**: The highest power is \( x^2 \) (from \( a^2 x^2 \)).
- **\( y \)**: The highest power is \( y^2 \) (from \( 6axy^2 \) and \( 9ay^2 \)).
- **\( 2 \)**: Appears in \( 6axy^2 \), so include \( 2 \) in the LCM.
- **\( 3 \)**: The highest power is \( 3^2 \) (from \( 9ay^2 \)).

### Step 3: Multiply the highest powers of all factors
The LCM is the product of the highest powers of all the factors:
\[
\text{LCM} = 2 \cdot 3^2 \cdot a^2 \cdot x^2 \cdot y^2
\]

### Step 4: Simplify the LCM
\[
\text{LCM} = 2 \cdot 9 \cdot a^2 \cdot x^2 \cdot y^2 = 18a^2x^2y^2
\]

Thus, the **LCM** of \( a^2 \times x^2 \), \( 6axy^2 \), and \( 9ay^2 \) is:
\[
\boxed{18a^2x^2y^2}
\]

Would you like further clarification or details on any of the steps?

### Related questions:
1. How do we find the greatest common divisor (GCD) of these terms?
2. What happens if a term had different exponents for \( a \), \( x \), or \( y \)?
3. Can you explain how the LCM changes if one of the terms is missing a variable?
4. How would the LCM change if the coefficients were different?
5. What is the LCM of two polynomials instead of these algebraic terms?

### Tip: 
When finding the LCM, always choose the highest power of each factor that appears across the terms.

Learn how to find the Least Common Multiple (LCM) of the algebraic expressions a^2×x^2, 6axy^2, and 9ay^2 with step-by-step factorization and detailed explanation. This problem covers key algebraic concepts such as factoring and the LCM formula, suitable for students in Grades 9-12.

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