Question

Find the greatest common factor for the list of monomials.
18 x cubed y Superscript 4 Baseline z​, 4 x squared y cubed​, 36 xy squared
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Part 1
The greatest common factor is
  
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Answer 1

To find the greatest common factor (GCF) of the monomials \( 18x^3y^4z \), \( 4x^2y^3 \), and \( 36xy^2 \), we need to determine the GCF of their coefficients and the GCF of the variable parts for each variable.

1. **Coefficients:**
- Coefficients: 18, 4, 36
- The GCF of 18, 4, and 36 can be calculated as follows:
- The prime factorization of each number is:
- \( 18 = 2 \times 3^2 \)
- \( 4 = 2^2 \)
- \( 36 = 2^2 \times 3^2 \)
- The common factors are \( 2^1 \) (which is the minimum power of 2 among the three) and no 3 since \( 4 \) does not have 3.
- Thus, \( GCF = 2^1 = 2 \).

2. **Variable Parts:**
- For \( x \):
- \( x^3 \) (from \( 18x^3y^4z \))
- \( x^2 \) (from \( 4x^2y^3 \))
- \( x^1 \) (from \( 36xy^2 \))
- The GCF of \( x^3, x^2, x^1 \) is \( x^1 = x \).

- For \( y \):
- \( y^4 \) (from \( 18x^3y^4z \))
- \( y^3 \) (from \( 4x^2y^3 \))
- \( y^2 \) (from \( 36xy^2 \))
- The GCF of \( y^4, y^3, y^2 \) is \( y^2 \).

- For \( z \):
- \( z^1 \) (from \( 18x^3y^4z \))
- \( z^0 \) (from \( 4x^2y^3 \))
- \( z^0 \) (from \( 36xy^2 \))
- Since \( z \) is missing in the last two monomials, the GCF is \( z^0 = 1 \) (not included).

3. **Combining the GCFs:**
- The overall GCF is the product of the GCFs of the coefficients and the variables:
\[
GCF = 2 \times x^1 \times y^2 = 2xy^2
\]

Thus, the greatest common factor for the list of monomials is:
\[
\boxed{2xy^2}
\]