To factor the expression [tex]\(45e - 27f\)[/tex] using the distributive property, follow these steps:
### Step 1: Identify the Greatest Common Factor (GCF)
First, find the greatest common factor (GCF) of the coefficients 45 and 27.
- The prime factorization of 45 is [tex]\(3^2 \times 5\)[/tex].
- The prime factorization of 27 is [tex]\(3^3\)[/tex].
The common prime factor in both numbers is [tex]\(3\)[/tex], and the highest power of 3 that appears in both factorizations is [tex]\(3^2\)[/tex]. Thus, the GCF of 45 and 27 is [tex]\(3^2 = 9\)[/tex].
### Step 2: Factor Out the GCF
Next, divide each term in the expression [tex]\(45e - 27f\)[/tex] by the GCF (which is 9) and use the distributive property to factor it out.
1. Divide [tex]\(45e\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
\frac{45e}{9} = 5e
\][/tex]
2. Divide [tex]\(27f\)[/tex] by [tex]\(9\)[/tex]:
[tex]\[
\frac{27f}{9} = 3f
\][/tex]
### Step 3: Write the Factored Expression
Now, rewrite the original expression [tex]\(45e - 27f\)[/tex] by factoring out the GCF:
[tex]\[
45e - 27f = 9(5e - 3f)
\][/tex]
So, the factored form of the expression [tex]\(45e - 27f\)[/tex] is:
[tex]\[
45e - 27f = 9(5e - 3f)
\][/tex]
This uses the distributive property to highlight the common factor and simplify the expression.
