Question

Identify the greatest common factor of 50cx^2y^2 and 25c^4w^3y^4

Answer 1

To find the greatest common factor (GCF) of the expressions \( 50cx^2y^2 \) and \( 25c^4w^3y^4 \), we can break down each term into its prime factors.

1. **Factor each coefficient:**

- The number 50 factors as \( 2 \times 5^2 \).
- The number 25 factors as \( 5^2 \).

2. **Find the GCF of the coefficients:**
- The GCF of 50 and 25 is 25, which factors as \( 5^2 \).

3. **Variables:**
- For \( c \):
- In \( 50cx^2y^2 \), the power of \( c \) is 1.
- In \( 25c^4w^3y^4 \), the power of \( c \) is 4.
- The GCF for \( c \) is \( c^{\min(1, 4)} = c^1 = c \).

- For \( x \):
- In \( 50cx^2y^2 \), the power of \( x \) is 2.
- In \( 25c^4w^3y^4 \), \( x \) is not present (power is 0).
- The GCF for \( x \) is \( x^{\min(2, 0)} = x^0 = 1 \) (not included).

- For \( y \):
- In \( 50cx^2y^2 \), the power of \( y \) is 2.
- In \( 25c^4w^3y^4 \), the power of \( y \) is 4.
- The GCF for \( y \) is \( y^{\min(2, 4)} = y^2 \).

- For \( w \):
- In \( 50cx^2y^2 \), \( w \) is not present (power is 0).
- In \( 25c^4w^3y^4 \), the power of \( w \) is 3.
- The GCF for \( w \) is \( w^{\min(0, 3)} = w^0 = 1 \) (not included).

4. **Combine the factors:**
- Coefficient: \( 25 \)
- \( c \): \( c^1 \)
- \( y \): \( y^2 \)

Thus, the GCF of \( 50cx^2y^2 \) and \( 25c^4w^3y^4 \) is:

\[
25cy^2
\]