whats the greatest common factor of 24x^4y^2 and 36xy^3

To find the greatest common factor (GCF) of \( 24x^4y^2 \) and \( 36xy^3 \), we break down both terms into their prime factorizations and powers of variables.

1. **Factor the coefficients:**
   - \( 24 = 2^3 \times 3 \)
   - \( 36 = 2^2 \times 3^2 \)

   The GCF of \( 24 \) and \( 36 \) is the smallest power of each prime factor:  
   - \( 2^2 \) (since the smallest power of 2 is \( 2^2 \))
   - \( 3^1 \) (since the smallest power of 3 is \( 3^1 \))

   Therefore, the GCF of the coefficients is \( 2^2 \times 3 = 12 \).

2. **Factor the variables:**
   - For \( x \), the smallest power is \( x^1 \) (from \( x^4 \) and \( x^1 \)).
   - For \( y \), the smallest power is \( y^2 \) (from \( y^2 \) and \( y^3 \)).

Thus, the GCF of the two terms is:

\[
\text{GCF} = 12x^1y^2 = 12xy^2
\]

### Final Answer: \( \boxed{12xy^2} \)

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How do you find the least common multiple (LCM) of two algebraic expressions?
2. How would you find the GCF of more than two algebraic terms?
3. What is the GCF of \( 48x^3y^4 \) and \( 72x^2y^5 \)?
4. How does the GCF change when both terms have negative coefficients?
5. How would you factor a polynomial using the GCF?

**Tip:** Always start by factoring both the numerical coefficients and the variables separately to find the GCF.

Learn how to find the greatest common factor (GCF) of algebraic terms like 24x^4y^2 and 36xy^3. This problem breaks down coefficients and variables using prime factorization. Suitable for students in Grades 7-9.

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