Certainly! Let's delve deeper into how the numerator is factored.

Original Expression

We start with the numerator:

$3^{n-4} - 3^{3n-3}$

Step 1: Identify the Common Factor

To factor the numerator, we need to find a common factor in both terms. Notice that both terms are powers of 3, specifically:

• The first term is $3^{n-4}$
• The second term is $3^{3n-3}$

The common factor between these two terms will be the term with the smaller exponent, which in this case is $3^{n-4}$.

Step 2: Factor Out the Common Term

Now, let's factor out $3^{n-4}$ from the numerator:

$3^{n-4} - 3^{3n-3} = 3^{n-4} \left( \frac{3^{n-4}}{3^{n-4}} - \frac{3^{3n-3}}{3^{n-4}} \right)$

Simplify the fractions inside the parentheses:

$3^{n-4} \left( 1 - 3^{3n-3-(n-4)} \right)$

Step 3: Simplify the Exponent

Now, simplify the exponent in the second term inside the parentheses:

$3n - 3 - (n - 4) = 3n - 3 - n + 4 = 2n + 1$

Thus, the expression inside the parentheses becomes:

$3^{n-4} \left(1 - 3^{2n+1}\right)$

Result

So, the numerator $3^{n-4} - 3^{3n-3}$ factors as:

$3^{n-4} \left(1 - 3^{2n+1}\right)$

This factorization allows you to cancel out the common term when simplifying the overall fraction.

Would you like further clarification on any part of this process?

Here are five related questions you might find helpful:

1. How do you determine the greatest common factor (GCF) of algebraic expressions?
2. Can you apply the distributive property to factor out terms in more complex expressions?
3. How does factoring help in solving algebraic equations?
4. What are some strategies for factoring differences of powers?
5. How can you verify that your factorization is correct?

Tip: When factoring expressions, always double-check the exponent operations—errors in simplifying exponents can lead to incorrect results.