To find the greatest common factor (GCF) of the expressions [tex]\( 12 w^5 v^3 x^2 \)[/tex] and [tex]\( 16 w^4 x^7 \)[/tex], follow these steps:
1. Identify the coefficients:
- The coefficient of the first expression, [tex]\( 12 w^5 v^3 x^2 \)[/tex], is 12.
- The coefficient of the second expression, [tex]\( 16 w^4 x^7 \)[/tex], is 16.
Find the greatest common divisor (GCD) of these coefficients:
- The prime factorization of 12 is [tex]\( 2^2 \times 3 \)[/tex].
- The prime factorization of 16 is [tex]\( 2^4 \)[/tex].
The GCD of 12 and 16 is [tex]\( 2^2 = 4 \)[/tex].
2. Determine the common variables with the lowest powers:
- For [tex]\( w \)[/tex]:
- The first expression has [tex]\( w^5 \)[/tex].
- The second expression has [tex]\( w^4 \)[/tex].
- The lowest power of [tex]\( w \)[/tex] common to both expressions is [tex]\( w^4 \)[/tex].
- For [tex]\( v \)[/tex]:
- The first expression has [tex]\( v^3 \)[/tex].
- The second expression does not contain [tex]\( v \)[/tex], hence the common factor in terms of [tex]\( v \)[/tex] is [tex]\( v^0 = 1 \)[/tex].
- For [tex]\( x \)[/tex]:
- The first expression has [tex]\( x^2 \)[/tex].
- The second expression has [tex]\( x^7 \)[/tex].
- The lowest power of [tex]\( x \)[/tex] common to both expressions is [tex]\( x^2 \)[/tex].
3. Combine the results:
- The GCD of the coefficients is 4.
- The common factors of the variables are [tex]\( w^4 \)[/tex] and [tex]\( x^2 \)[/tex].
Thus, the greatest common factor (GCF) of the expressions [tex]\( 12 w^5 v^3 x^2 \)[/tex] and [tex]\( 16 w^4 x^7 \)[/tex] is:
[tex]\[
4 w^4 x^2
\][/tex]
