If `(9^(n)xx3^(2)xx(3^(-n//2))^(-2)-(27)^(n))/(3^(3m)xx2^(3))=(1)/(27)` , then find m-n .

Answer

To solve the equation

\[ \frac{9^n \cdot 3^2 \cdot (3^{-n/2})^{-2} - 27^n}{3^{3m} \cdot 2^3} = \frac{1}{27}, \]

we will simplify step by step.

### Step 1: Rewrite the terms in the equation

First, we can express \(9\) and \(27\) in terms of base \(3\):

- \(9 = 3^2\), so \(9^n = (3^2)^n = 3^{2n}\).
- \(27 = 3^3\), so \(27^n = (3^3)^n = 3^{3n}\).

Now, substituting these into the equation gives:

\[ \frac{3^{2n} \cdot 3^2 \cdot (3^{-n/2})^{-2} - 3^{3n}}{3^{3m} \cdot 2^3} = \frac{1}{27}. \]

### Step 2: Simplify the expression

Next, simplify the term \((3^{-n/2})^{-2}\):

\[ (3^{-n/2})^{-2} = 3^{n}. \]

Now substituting this back into the equation gives:

\[ \frac{3^{2n} \cdot 3^2 \cdot 3^n - 3^{3n}}{3^{3m} \cdot 2^3} = \frac{1}{27}. \]

### Step 3: Combine the powers of \(3\)

Combine the powers of \(3\) in the numerator:

\[ 3^{2n} \cdot 3^2 \cdot 3^n = 3^{2n + 2 + n} = 3^{3n + 2}. \]

So, the equation now looks like:

\[ \frac{3^{3n + 2} - 3^{3n}}{3^{3m} \cdot 2^3} = \frac{1}{27}. \]

### Step 4: Factor the numerator

Factor out \(3^{3n}\) from the numerator:

\[ 3^{3n}(3^2 - 1) = 3^{3n}(9 - 1) = 3^{3n} \cdot 8. \]

Thus, the equation simplifies to:

\[ \frac{3^{3n} \cdot 8}{3^{3m} \cdot 2^3} = \frac{1}{27}. \]

### Step 5: Simplify further

Since \(2^3 = 8\), we can rewrite the equation as:

\[ \frac{3^{3n} \cdot 8}{3^{3m} \cdot 8} = \frac{1}{27}. \]

This simplifies to:

\[ \frac{3^{3n}}{3^{3m}} = \frac{1}{27}. \]

### Step 6: Rewrite \(27\) in terms of \(3\)

Since \(27 = 3^3\), we can rewrite the equation as:

\[ 3^{3n - 3m} = 3^{-3}. \]

### Step 7: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other:

\[ 3n - 3m = -3. \]

Dividing the entire equation by \(3\):

\[ n - m = -1. \]

### Step 8: Solve for \(m\)

Rearranging gives:

\[ m = n + 1. \]

### Step 9: Find \(m - n\)

Now, we can find \(m - n\):

\[ m - n = (n + 1) - n = 1. \]

Thus, the final answer is:

\[ \boxed{1}. \]