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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!

Simplify the expression: (8^9 * 3^6) / (3^5 * 8^15) ÷ (1/8)

Learn how to simplify the expression (8^9 * 3^6) / (3^5 * 8^15) ÷ (1/8) step-by-step. This problem covers exponent rules and division of fractions. Suitable for students in Grades 9-10.

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