If $2^{3m}\times 8 = 2^9 \: and \: 3^{2n} \times 9 = 3{6}$ , then find the value $\frac{m}{n}$.

We are given:

$2^{3m} \times 8 = 2^9 \quad \text{and} \quad 3^{2n} \times 9 = 3^6$

Step 1: Express numbers as powers of the same base

$8 = 2^3 \implies 2^{3m} \cdot 2^3 = 2^9$

$9 = 3^2 \implies 3^{2n} \cdot 3^2 = 3^6$

Step 2: Combine exponents

For the first equation:

$2^{3m+3} = 2^9 \implies 3m + 3 = 9 \implies 3m = 6 \implies m = 2$

For the second equation:

$3^{2n+2} = 3^6 \implies 2n + 2 = 6 \implies 2n = 4 \implies n = 2$

Step 3: Find $\frac{m}{n}$

$\frac{m}{n} = \frac{2}{2} = 1$

✅ Answer:

$\frac{m}{n} = 1$