To factor the greatest common factor (GCF) out of the expression [tex]\(18a^4b^3 + 45a^2b^5\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients are 18 and 45.
- The greatest common factor (GCF) of 18 and 45 is 9.
2. Identify the GCF of the variables:
- For the [tex]\(a\)[/tex] terms, we have [tex]\(a^4\)[/tex] and [tex]\(a^2\)[/tex].
- The GCF of [tex]\(a^4\)[/tex] and [tex]\(a^2\)[/tex] is [tex]\(a^2\)[/tex] (the variable raised to the lowest power).
- For the [tex]\(b\)[/tex] terms, we have [tex]\(b^3\)[/tex] and [tex]\(b^5\)[/tex].
- The GCF of [tex]\(b^3\)[/tex] and [tex]\(b^5\)[/tex] is [tex]\(b^3\)[/tex] (the variable raised to the lowest power).
Combining these, the GCF for the variables is [tex]\(a^2b^3\)[/tex].
3. Combine the GCF of the coefficients and the variables:
- The full GCF of the expression is [tex]\(9a^2b^3\)[/tex].
4. Factor the GCF out of the expression:
- To factor [tex]\(9a^2b^3\)[/tex] out of [tex]\(18a^4b^3\)[/tex]:
- Divide [tex]\(18a^4b^3\)[/tex] by [tex]\(9a^2b^3\)[/tex]:
[tex]\[
\frac{18a^4b^3}{9a^2b^3} = 2a^2
\][/tex]
- To factor [tex]\(9a^2b^3\)[/tex] out of [tex]\(45a^2b^5\)[/tex]:
- Divide [tex]\(45a^2b^5\)[/tex] by [tex]\(9a^2b^3\)[/tex]:
[tex]\[
\frac{45a^2b^5}{9a^2b^3} = 5b^2
\][/tex]
5. Write the factored expression:
- The factored form of the expression is:
[tex]\[
9a^2b^3 (2a^2 + 5b^2)
\][/tex]
Thus, the answer is:
[tex]\[
9a^2b^3 \cdot (2a^2 + 5b^2)
\][/tex]
