Find the value of `(3^(12+n)xx9^(2n-7))/(3^(5n))`

Answer

To solve the expression \(\frac{3^{12+n} \times 9^{2n-7}}{3^{5n}}\), we will follow these steps:

### Step 1: Rewrite \(9^{2n-7}\) in terms of base 3
Since \(9\) can be expressed as \(3^2\), we can rewrite \(9^{2n-7}\) as:
\[ 9^{2n-7} = (3^2)^{2n-7} = 3^{2(2n-7)} = 3^{4n - 14} \]

### Step 2: Substitute back into the expression
Now, substitute \(9^{2n-7}\) back into the original expression:
\[ \frac{3^{12+n} \times 3^{4n-14}}{3^{5n}} \]

### Step 3: Combine the powers of 3 in the numerator
Using the property of exponents \(a^m \times a^n = a^{m+n}\), we can combine the powers in the numerator:
\[ 3^{12+n + 4n - 14} = 3^{(12 - 14) + (n + 4n)} = 3^{-2 + 5n} = 3^{5n - 2} \]

### Step 4: Simplify the expression
Now, we have:
\[ \frac{3^{5n-2}}{3^{5n}} \]
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify this to:
\[ 3^{(5n - 2) - 5n} = 3^{-2} \]

### Step 5: Final simplification
We can rewrite \(3^{-2}\) as:
\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]

Thus, the value of the expression is:
\[ \frac{1}{9} \]