To find the greatest common factor (GCF) of the two expressions [tex]\(18 w^8 y^5 u^6\)[/tex] and [tex]\(24 w^7 y^4\)[/tex], we will consider the coefficients and variables separately.
Step 1: Find the GCF of the coefficients:
The coefficients are 18 and 24. To find the GCF, we can list the factors of each number.
- Factors of 18: [tex]\(1, 2, 3, 6, 9, 18\)[/tex]
- Factors of 24: [tex]\(1, 2, 3, 4, 6, 8, 12, 24\)[/tex]
The greatest common factor of 18 and 24 is 6.
Step 2: Find the GCF of the variables [tex]\(w^8\)[/tex] and [tex]\(w^7\)[/tex]:
The base variable is [tex]\(w\)[/tex]. To find the GCF, we take the lowest power of [tex]\(w\)[/tex] that appears in both expressions.
- In [tex]\(w^8\)[/tex], the power of [tex]\(w\)[/tex] is 8.
- In [tex]\(w^7\)[/tex], the power of [tex]\(w\)[/tex] is 7.
The lowest power of [tex]\(w\)[/tex] common to both terms is [tex]\(w^7\)[/tex].
Step 3: Find the GCF of the variables [tex]\(y^5\)[/tex] and [tex]\(y^4\)[/tex]:
The base variable is [tex]\(y\)[/tex]. To find the GCF, we take the lowest power of [tex]\(y\)[/tex] that appears in both expressions.
- In [tex]\(y^5\)[/tex], the power of [tex]\(y\)[/tex] is 5.
- In [tex]\(y^4\)[/tex], the power of [tex]\(y\)[/tex] is 4.
The lowest power of [tex]\(y\)[/tex] common to both terms is [tex]\(y^4\)[/tex].
Step 4: Find the GCF of the variable [tex]\(u^6\)[/tex]:
The variable [tex]\(u\)[/tex] appears only in the first expression ([tex]\(u^6\)[/tex]) and not in the second expression. Since [tex]\(u\)[/tex] does not appear in both expressions, it does not contribute to the GCF.
Combining everything together:
The GCF of the coefficients is 6.
The GCF of the variable [tex]\(w\)[/tex] is [tex]\(w^7\)[/tex].
The GCF of the variable [tex]\(y\)[/tex] is [tex]\(y^4\)[/tex].
Thus, the GCF of the expressions [tex]\(18 w^8 y^5 u^6\)[/tex] and [tex]\(24 w^7 y^4\)[/tex] is:
[tex]\[
6 w^7 y^4
\][/tex]
This is the greatest common factor of the given expressions.
