To simplify the expression \(\frac{3^{3n} \times 9^{4n}}{9^{3n} + 3}\), let's follow the steps below:

### Step 1: Simplify \(9^{4n}\) and \(9^{3n}\)
Since \(9\) can be written as \(3^2\), we can express the powers of 9 in terms of base 3:

\[
9^{4n} = (3^2)^{4n} = 3^{8n}
\]
\[
9^{3n} = (3^2)^{3n} = 3^{6n}
\]

### Step 2: Substitute back into the expression
Now, replace \(9^{4n}\) and \(9^{3n}\) with their simplified forms:

\[
\frac{3^{3n} \times 3^{8n}}{3^{6n} + 3}
\]

### Step 3: Combine exponents in the numerator
Since the bases are the same in the numerator, we can add the exponents:

\[
3^{3n} \times 3^{8n} = 3^{(3n + 8n)} = 3^{11n}
\]

So, the expression becomes:

\[
\frac{3^{11n}}{3^{6n} + 3}
\]

### Step 4: Factor the denominator
Factor out the common factor in the denominator:

\[
3^{6n} + 3 = 3(3^{6n-1} + 1)
\]

So, the expression simplifies to:

\[
\frac{3^{11n}}{3(3^{6n-1} + 1)} = \frac{3^{11n-1}}{3^{6n-1} + 1}
\]

This is the simplified form of the given expression.

### Summary
The expression \(\frac{3^{3n} \times 9^{4n}}{9^{3n} + 3}\) simplifies to:

\[
\frac{3^{11n-1}}{3^{6n-1} + 1}
\]

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

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Here are five related questions:
1. How would the expression change if the exponent of \(n\) was halved?
2. What happens to the expression if \(n = 0\)?
3. How do the properties of exponents simplify calculations involving powers of 9?
4. Can you simplify \(3^{3n} \times 27^{2n}\) similarly?
5. What if the base in the expression was different, say \(4\) instead of \(3\)?

**Tip:** Remember that factoring common terms is often a key step in simplifying algebraic expressions.

Learn how to simplify the expression 3^(3n) x 9^(4n) / (9^(3n) + 3) step-by-step. This problem involves exponents and factorization. Suitable for high school students and above.

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