Let's find the greatest common factor (GCF) of the two terms [tex]\(40x^5 y^2\)[/tex] and [tex]\(32x^2 y^3\)[/tex].
### Step 1: Determine the GCF of the coefficients
The coefficients are 40 and 32. To find the GCF of these two numbers:
- The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.
- The factors of 32 are 1, 2, 4, 8, 16, and 32.
The greatest common factor of 40 and 32 is the largest number that appears in both lists of factors. This number is 8.
### Step 2: Determine the minimum exponents of [tex]\(x\)[/tex]
The terms contain the variables [tex]\(x\)[/tex] raised to different powers:
- In the first term, [tex]\(x\)[/tex] is raised to the 5th power ([tex]\(x^5\)[/tex]).
- In the second term, [tex]\(x\)[/tex] is raised to the 2nd power ([tex]\(x^2\)[/tex]).
For the GCF, we take the lower exponent of [tex]\(x\)[/tex], which is 2.
### Step 3: Determine the minimum exponents of [tex]\(y\)[/tex]
The terms contain the variables [tex]\(y\)[/tex] raised to different powers:
- In the first term, [tex]\(y\)[/tex] is raised to the 2nd power ([tex]\(y^2\)[/tex]).
- In the second term, [tex]\(y\)[/tex] is raised to the 3rd power ([tex]\(y^3\)[/tex]).
For the GCF, we take the lower exponent of [tex]\(y\)[/tex], which is 2.
### Step 4: Combine the results
Now, combine the coefficient, exponents of [tex]\(x\)[/tex], and exponents of [tex]\(y\)[/tex] to form the GCF of the terms.
The GCF of [tex]\(40x^5 y^2\)[/tex] and [tex]\(32x^2 y^3\)[/tex] is:
[tex]\[ 8x^2y^2 \][/tex]
### Final Answer
So, the greatest common factor of [tex]\(40 x^5 y^2\)[/tex] and [tex]\(32 x^2 y^3\)[/tex] is [tex]\(8x^2y^2\)[/tex].
