Question

Find the greatest common factor for the list of monomials.
x Superscript 5 Baseline y Superscript 5 Baseline z Superscript 5​, y cubed z Superscript 5​, xy cubed z squared
Question content area bottom
Part 1
The GCF is
  
enter your response here.
​(Simplify your​ answer.)

Answer 1

To find the greatest common factor (GCF) of the given monomials \( x^5 y^5 z^5 \), \( y^3 z^5 \), and \( xy^3 z^2 \), we will identify the lowest powers of each variable present in all monomials.

1. **Identify the powers of each variable** in the monomials:

- **Monomial 1:** \( x^5 y^5 z^5 \)
- \( x: 5 \)
- \( y: 5 \)
- \( z: 5 \)

- **Monomial 2:** \( y^3 z^5 \)
- \( x: 0 \)
- \( y: 3 \)
- \( z: 5 \)

- **Monomial 3:** \( xy^3 z^2 \)
- \( x: 1 \)
- \( y: 3 \)
- \( z: 2 \)

2. **Find the GCF for each variable** by taking the lowest power of each variable that appears in all monomials:

- For \( x \): the lowest power is \( \min(5, 0, 1) = 0 \). Therefore, \( x^0 = 1 \).
- For \( y \): the lowest power is \( \min(5, 3, 3) = 3 \). Therefore, \( y^3 \).
- For \( z \): the lowest power is \( \min(5, 5, 2) = 2 \). Therefore, \( z^2 \).

3. **Combine the GCF of each variable** to write the overall GCF:

\[
\text{GCF} = 1 \cdot y^3 \cdot z^2 = y^3 z^2
\]

So, the greatest common factor of the given monomials is

\[
\boxed{y^3 z^2}
\]