The expression provided is:

$\frac{8^9 \cdot 3^6}{3^5 \cdot 8^{15}} \div \frac{1}{8}$

Let's simplify this step by step:

Step 1: Simplify the Fraction

We can handle the numerator and denominator separately:

$\frac{8^9 \cdot 3^6}{3^5 \cdot 8^{15}} = \frac{8^9}{8^{15}} \cdot \frac{3^6}{3^5}$

For powers with the same base, we subtract the exponents:

$\frac{8^9}{8^{15}} = 8^{9-15} = 8^{-6}$ $\frac{3^6}{3^5} = 3^{6-5} = 3^1 = 3$

So the fraction simplifies to:

$8^{-6} \cdot 3$

Step 2: Handle the Division

The original problem also has a division by $\frac{1}{8}$. Dividing by a fraction is the same as multiplying by its reciprocal, so:

$8^{-6} \cdot 3 \div \frac{1}{8} = 8^{-6} \cdot 3 \cdot 8 = 8^{-6+1} \cdot 3 = 8^{-5} \cdot 3$

Step 3: Final Simplified Expression

The final simplified expression is:

$3 \cdot 8^{-5}$

This can also be written as:

$\frac{3}{8^5}$

Thus, the simplified form of the given expression is:

$\boxed{\frac{3}{8^5}}$

Would you like more details or have any other questions?

Here are 5 related questions for practice:

1. Simplify $\frac{2^8 \cdot 5^4}{5^2 \cdot 2^3}$.
2. What is the result of $\frac{9^3 \cdot 4}{4 \cdot 9^2}$?
3. Simplify $\frac{7^5 \cdot 5}{5 \cdot 7^2}$.
4. Calculate $\frac{1}{2^4} \cdot 2^3$.
5. Simplify $\frac{6^8 \cdot 3^5}{6^3 \cdot 3^2}$.

Tip: When dividing exponents with the same base, subtract the powers!