simplify $$ \frac{3^{n-4}-3^{3n-3}}{3^{4n+4}} $$

To simplify the expression 

$$ \frac{3^{n-4}-3^{3n-3}}{3^{4n+4}} $$

we can start by factoring the numerator and simplifying the entire expression step by step:

### Step 1: Factor the numerator
The numerator is $ 3^{n-4} - 3^{3n-3} $. Notice that both terms share a common factor of $ 3^{n-4} $. We can factor this out:

\[
3^{n-4} - 3^{3n-3} = 3^{n-4} \left(1 - 3^{3n-3-(n-4)}\right)
\]

Simplify the exponent inside the parentheses:

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

So the numerator becomes:

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

### Step 2: Simplify the fraction
Now, substitute the simplified numerator back into the original fraction:

\[
\frac{3^{n-4} \left(1 - 3^{2n+1}\right)}{3^{4n+4}}
\]

This expression can be further simplified by canceling out the common term $ 3^{n-4} $ from the numerator with part of the denominator:

\[
\frac{1 - 3^{2n+1}}{3^{4n+4-(n-4)}}
\]

Simplify the exponent in the denominator:

\[
\frac{1 - 3^{2n+1}}{3^{3n+8}}
\]

### Final Answer
So, the simplified expression is:

\[
\frac{1 - 3^{2n+1}}{3^{3n+8}}
\]

Would you like more details on this solution, or do you have any questions?

Here are five related questions you might find helpful:

1. How do you factor common terms out of an algebraic expression?
2. What are the properties of exponents that can help simplify expressions like this?
3. How can you simplify fractions with exponents in the numerator and denominator?
4. What happens when you factor terms in an expression with more than one variable?
5. How do exponent rules apply when dividing terms with different exponents?

**Tip:** Always look for common factors when simplifying algebraic fractions—it can greatly reduce the complexity of the expression.

Learn how to simplify the fraction $ \frac{3^{n-4} - 3^{3n-3}}{3^{4n+4}} $ with detailed step-by-step solution. This problem involves exponents and fraction simplification techniques, suitable for high school students.

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