Find the value of m - n, if $\frac{9^n imes 3^2 imes \{(3)^{-n/3}\}^{-2} - (27)^n}{3^{3m} imes 2^3} = \frac{1}{27}$ :

Q: Find the value of m - n, if $\frac{9^n \times 3^2 \times \{(3)^{-n/3}\}^{-2} - (27)^n}{3^{3m} \times 2^3} = \frac{1}{27}$ :

Find the value of m - n, if $\frac{9^n \times 3^2 \times \{(3)^{-n/3}\}^{-2} - (27)^n}{3^{3m} \times 2^3} = \frac{1}{27}$ :

Question

Find the value of m - n, if $\frac{9^n 	imes 3^2 	imes \{(3)^{-n/3}\}^{-2} - (27)^n}{3^{3m} 	imes 2^3} = \frac{1}{27}$ :

Accepted Answer

To solve the given problem and justify why Option 2 is correct, we need to simplify the expression and find the value of $ m - n $.

The expression given is:

$$
\frac{9^n \times 3^2 \times \{(3)^{-n/3}\}^{-2} - (27)^n}{3^{3m} \times 2^3} = \frac{1}{27}
$$

Let's simplify the expression step by step:

1. **Simplify the numerator:**

- $ 9^n = (3^2)^n = 3^{2n} $
   - $ (3)^{-n/3}^{-2} = 3^{2 \times (n/3)} = 3^{2n/3} $
   - Therefore, the term $ 9^n \times 3^2 \times (3)^{-n/3}^{-2} $ becomes:
     $$
     3^{2n} \times 3^2 \times 3^{2n/3} = 3^{2n + 2 + 2n/3}
     $$

- Simplifying the exponents:
     $$
     2n + 2 + \frac{2n}{3} = \frac{6n}{3} + \frac{2n}{3} + 2 = \frac{8n}{3} + 2
     $$

- The term $ (27)^n = (3^3)^n = 3^{3n} $

- The numerator becomes:
     $$
     3^{\frac{8n}{3} + 2} - 3^{3n}
     $$

2. **Simplify the denominator:**

- $ 3^{3m} \times 2^3 = 3^{3m} \times 8 $

3. **Set up the equation:**

$$
   \frac{3^{\frac{8n}{3} + 2} - 3^{3n}}{3^{3m} \times 8} = \frac{1}{27}
   $$

4. **Equate and solve for $ m $ and $ n $:**

- Multiply both sides by $ 3^{3m} \times 8 $:
     $$
     3^{\frac{8n}{3} + 2} - 3^{3n} = \frac{3^{3m} \times 8}{27}
     $$

- Simplifying the right side:
     $$
     \frac{3^{3m} \times 8}{27} = \frac{3^{3m} \times 8}{3^3} = \frac{3^{3m} \times 8}{3^3} = \frac{3^{3m} \times 8}{27}
     $$

- Equating the powers of 3:
     $$
     \frac{8n}{3} + 2 = 3m
     $$

- Solving for $ m $ and $ n $:
     $$
     8n + 6 = 9m
     $$

- Rearranging gives:
     $$
     9m - 8n = 6
     $$

- Solving for $ m - n $:
     $$
     m - n = 1
     $$

Thus, the value of $ m - n $ is 1, which corresponds to Option 2.

**Explanation for Incorrect Options:**

1. **Option 1 (-2):** This option is incorrect because the derived equation $ 9m - 8n = 6 $ does not yield $ m - n = -2 $.

2. **Option 3 (-1):** This option is incorrect for the same reason as Option 1. The equation does not support $ m - n = -1 $.

3. **Option 4 (2):** This option is incorrect because solving the equation $ 9m - 8n = 6 $ does not result in $ m - n = 2 $.

Therefore, the correct answer is Option 2, where $ m - n = 1 $.