Question

How would you rewrite the expression 24ab2c+18a3b
by factoring out the GCF?

Answer 1

To factor out the greatest common factor (GCF) from the expression \( 24ab^2c + 18a^3b \), we first need to determine the GCF of the coefficients and the variables.

1. **Find the GCF of the coefficients (24 and 18):**
- The prime factorization of 24 is \( 2^3 \times 3 \).
- The prime factorization of 18 is \( 2 \times 3^2 \).
- The GCF of 24 and 18 is obtained by taking the lowest power of each prime factor present in both factorizations:
- For \( 2 \), the minimum power is \( 2^1 = 2 \).
- For \( 3 \), the minimum power is \( 3^1 = 3 \).
- Thus, the GCF of 24 and 18 is \( 2^1 \times 3^1 = 6 \).

2. **Find the GCF of the variables:**
- For \( ab^2c \) and \( a^3b \):
- The lowest power of \( a \) is \( a^1 \) (since \( b^2c \) has \( a^1 \) and \( a^3b \) has \( a^3 \)).
- The lowest power of \( b \) is \( b^1 \) (the first term has \( b^2 \) and the second has \( b^1 \)).
- The variable \( c \) appears only in the first term, so we do not include it in the GCF.
- Therefore, the GCF of the variables is \( ab \).

3. **Combine the GCF of coefficients and the variables:**
- The overall GCF of \( 24ab^2c + 18a^3b \) is \( 6ab \).

4. **Factor out the GCF from the expression:**
- Now we can rewrite the expression:
\[
24ab^2c + 18a^3b = 6ab(4bc + 3a^2)
\]

Thus, the expression \( 24ab^2c + 18a^3b \) can be factored as:
\[
6ab(4bc + 3a^2)
\]