$$ \frac{3^{3 n-1} \cdot 9^{n} \cdot 3}{9^{3 n}} \quad $$ Simplify

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{3^{3n - 1} \cdot 9^{n} \cdot 3}{9^{3n}}
$$

Let's express everything in terms of the base $$3$$:

1. **Convert $$9$$ to $$3^2$$:**
   $$
   9^n = (3^2)^n = 3^{2n} \quad \text{and} \quad 9^{3n} = (3^2)^{3n} = 3^{6n}
   $$

2. **Rewrite the original expression using these conversions:**
   $$
   \frac{3^{3n - 1} \cdot 3^{2n} \cdot 3}{3^{6n}}
   $$

3. **Combine the exponents in the numerator:**
   $$
   3^{3n - 1 + 2n + 1} = 3^{5n}
   $$
   (Here, $$-1 + 1$$ cancels out.)

4. **Divide the exponents by subtracting the denominator's exponent:**
   $$
   \frac{3^{5n}}{3^{6n}} = 3^{5n - 6n} = 3^{-n}
   $$

So, the simplified expression is:

$$
3^{-n} \quad \text{or} \quad \frac{1}{3^n}
$$

**Final Answer:**
$$
3^{-n}
$$
The simplified expression is $$3^{-n}$$ or $$\frac{1}{3^n}$$.

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