Find the GCF of the following expressions. (Assume x and y are integers and that n is nonzero.)
2x6ny7n, 10x9ny6n

To find the greatest common factor (GCF) of the expressions \(2x^{6n}y^{7n}\) and \(10x^{9n}y^{6n}\), we need to follow these steps:

1. **Factor the coefficients**:  
   - The coefficients are 2 and 10.
   - The GCF of 2 and 10 is 2.

2. **Find the GCF of the \(x\)-terms**:  
   - The exponents of \(x\) in the first and second terms are \(6n\) and \(9n\), respectively.
   - The GCF of \(x^{6n}\) and \(x^{9n}\) is \(x^{\text{min}(6n, 9n)} = x^{6n}\).

3. **Find the GCF of the \(y\)-terms**:  
   - The exponents of \(y\) in the first and second terms are \(7n\) and \(6n\), respectively.
   - The GCF of \(y^{7n}\) and \(y^{6n}\) is \(y^{\text{min}(7n, 6n)} = y^{6n}\).

4. **Multiply the GCFs together**:
   - Combine the GCFs of the coefficients, \(x\)-terms, and \(y\)-terms.

\[
\text{GCF} = 2 \cdot x^{6n} \cdot y^{6n} = 2x^{6n}y^{6n}
\]

So, the GCF of the expressions \(2x^{6n}y^{7n}\) and \(10x^{9n}y^{6n}\) is \(2x^{6n}y^{6n}\).

Would you like to know more details or have any questions? Here are some related problems you might find interesting:

1. Find the GCF of \(15x^{5n}y^{8n}\) and \(25x^{7n}y^{6n}\).
2. Determine the LCM of \(2x^{6n}y^{7n}\) and \(10x^{9n}y^{6n}\).
3. Simplify the expression \(\frac{2x^{6n}y^{7n}}{10x^{9n}y^{6n}}\).
4. If \(n = 1\), what is the GCF of the expressions?
5. How does changing the value of \(n\) affect the GCF?

**Tip:** When finding the GCF of algebraic expressions, always focus on the smallest exponents for each variable and the GCF of the coefficients.

Learn how to find the greatest common factor (GCF) of the expressions 2x^6ny^7n and 10x^9ny^6n with a detailed step-by-step solution. This problem covers algebraic concepts, including polynomials and GCF calculations for algebraic expressions. Suitable for high school students.

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