Simplify :
`(3xx9^(n+1)-9xx3^(2n))/(3xx3^(2n+3)-9^(n+1))`

Answer

To simplify the expression \((3 \cdot 9^{(n+1)} - 9 \cdot 3^{(2n)}) / (3 \cdot 3^{(2n+3)} - 9^{(n+1)})\), we will follow these steps:

### Step 1: Rewrite the terms using powers of 3
We know that \(9 = 3^2\). Therefore, we can rewrite \(9^{(n+1)}\) as \((3^2)^{(n+1)} = 3^{2(n+1)} = 3^{(2n + 2)}\).

So, the expression becomes:
\[ \frac{3 \cdot 3^{(2n + 2)} - 9 \cdot 3^{(2n)}}{3 \cdot 3^{(2n + 3)} - 3^{(2(n+1))}} \]

### Step 2: Simplify the numerator
Now, we can simplify the numerator:
\[ 3 \cdot 3^{(2n + 2)} - 9 \cdot 3^{(2n)} = 3^{(1 + 2n + 2)} - 9 \cdot 3^{(2n)} = 3^{(2n + 3)} - 3^{(2)} \cdot 3^{(2n)} = 3^{(2n + 3)} - 3^{(2n + 2)} \]

### Step 3: Factor out common terms in the numerator
The numerator can be factored:
\[ 3^{(2n + 2)}(3 - 1) = 3^{(2n + 2)} \cdot 2 \]

### Step 4: Simplify the denominator
Now, let's simplify the denominator:
\[ 3 \cdot 3^{(2n + 3)} - 9^{(n + 1)} = 3 \cdot 3^{(2n + 3)} - 3^{(2(n + 1))} = 3^{(1 + 2n + 3)} - 3^{(2n + 2)} = 3^{(2n + 4)} - 3^{(2n + 2)} \]

### Step 5: Factor out common terms in the denominator
The denominator can be factored:
\[ 3^{(2n + 2)}(3^2 - 1) = 3^{(2n + 2)}(9 - 1) = 3^{(2n + 2)} \cdot 8 \]

### Step 6: Combine the simplified numerator and denominator
Now, substituting back into the expression, we have:
\[ \frac{3^{(2n + 2)} \cdot 2}{3^{(2n + 2)} \cdot 8} \]

### Step 7: Cancel out the common terms
The \(3^{(2n + 2)}\) terms cancel out:
\[ \frac{2}{8} = \frac{1}{4} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{1}{4} \]

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