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Let's analyze and find the greatest common factor (GCF) of the given expressions:

The expressions are:
1. \( 6x^{11}y^2z^8 \)
2. \( 12x^8y^2z^{14} \)
3. \( 2xyz^9 \)

### Step-by-Step Solution:

1. **Numerical Coefficients**:
   - The coefficients are 6, 12, and 2.
   - The GCF of 6, 12, and 2 is **2**.

2. **For the variable \( x \)**:
   - The powers of \( x \) are 11, 8, and 1.
   - The GCF is the lowest power of \( x \), which is **1**.
   - So, the GCF of the \( x \)-terms is \( x^1 \) or simply \( x \).

3. **For the variable \( y \)**:
   - The powers of \( y \) are 2, 2, and 1.
   - The GCF is the lowest power of \( y \), which is **1**.
   - So, the GCF of the \( y \)-terms is \( y^1 \) or simply \( y \).

4. **For the variable \( z \)**:
   - The powers of \( z \) are 8, 14, and 9.
   - The GCF is the lowest power of \( z \), which is **8**.
   - So, the GCF of the \( z \)-terms is \( z^8 \).

### Final GCF:
\[
\boxed{2xyz^8}
\]

This is the greatest common factor of the three expressions.

Would you like further clarification or details on any step?

Here are 5 related questions you might find interesting:
1. How do you find the GCF of three numbers without variables?
2. What is the method to find the least common multiple (LCM) of algebraic expressions?
3. Can the GCF of two or more expressions be a polynomial?
4. How do you factor a polynomial using the GCF?
5. How does the GCF apply when simplifying fractions?

**Tip:** When finding the GCF of algebraic expressions, always consider both the numerical coefficients and the variables separately, using the lowest exponent for each variable.

Find the GCF of the expressions: 6x^11 y^2 z^8, 12x^8 y^2 z^14, and 2xyz^9.

Learn how to find the greatest common factor (GCF) of algebraic expressions with variables and exponents. This problem explores the GCF of the expressions 6x^11 y^2 z^8, 12x^8 y^2 z^14, and 2xyz^9 with a step-by-step solution, covering key concepts like exponents and factorization.

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