Sure! Let's work through the problem of simplifying the expression [tex]\(18 p^8 q^3 + 28 p^6 q^5\)[/tex] by factoring out the greatest common factor.
### Step-by-Step Solution:
1. Identify the coefficients and find their greatest common factor (GCF):
- The coefficients are 18 and 28.
- The prime factorization of 18 is [tex]\(18 = 2 \times 3^2\)[/tex].
- The prime factorization of 28 is [tex]\(28 = 2 \times 2 \times 7 = 2^2 \times 7\)[/tex].
- The common factor between 18 and 28 is 2.
- Therefore, the GCF of the coefficients is 2.
2. Identify the common factors of the variables:
- For [tex]\(p^8\)[/tex] and [tex]\(p^6\)[/tex], the common factor is [tex]\(p^6\)[/tex] (the lowest power of [tex]\(p\)[/tex] common to both terms).
- For [tex]\(q^3\)[/tex] and [tex]\(q^5\)[/tex], the common factor is [tex]\(q^3\)[/tex] (the lowest power of [tex]\(q\)[/tex] common to both terms).
3. Combine the GCF of the coefficients and the variables:
- The overall common factor of the entire expression is the product of the GCF of the coefficients and the common factors of the variables.
- This results in a common factor of [tex]\(2 p^6 q^3\)[/tex].
4. Factor out the common factor from the original expression:
- Original expression: [tex]\(18 p^8 q^3 + 28 p^6 q^5\)[/tex]
- Factored form:
[tex]\[
18 p^8 q^3 + 28 p^6 q^5 = 2 p^6 q^3 \left( 9 p^2 + 14 q^2 \right)
\][/tex]
Here's the detailed factoring process:
- Divide each term by the common factor [tex]\(2 p^6 q^3\)[/tex]:
- [tex]\( \frac{18 p^8 q^3}{2 p^6 q^3} = 9 p^2 \)[/tex]
- [tex]\( \frac{28 p^6 q^5}{2 p^6 q^3} = 14 q^2 \)[/tex]
Therefore, the expression [tex]\(18 p^8 q^3 + 28 p^6 q^5\)[/tex] can be factored as:
[tex]\[ 18 p^8 q^3 + 28 p^6 q^5 = 2 p^6 q^3 (9 p^2 + 14 q^2) \][/tex]
This is the simplified form of the given expression.
