The given expression is:

$\frac{6m^3}{(3m^3)^3}$

Let's solve this step-by-step.

Step 1: Simplify the denominator

The denominator is $(3m^3)^3$. We can expand this:

$(3m^3)^3 = 3^3 \cdot (m^3)^3 = 27m^9$

Step 2: Rewrite the entire expression

Now the expression becomes:

$\frac{6m^3}{27m^9}$

Step 3: Simplify the fraction

We can simplify the numerical part and the powers of $m$:

$\frac{6}{27} = \frac{2}{9}$

For the powers of $m$, apply the rule for dividing exponents: $m^a / m^b = m^{a-b}$:

$m^{3-9} = m^{-6}$

Step 4: Combine the results

Now the expression is:

$\frac{2}{9}m^{-6}$

This can also be written as:

$\frac{2}{9m^6}$

Final Answer:

$\frac{2}{9m^6}$

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How would the result change if the powers of $m$ were different?
2. What happens if the base numbers (like 6 and 3) are not constants?
3. How can you apply this method to simplify expressions involving multiple variables?
4. What is the general rule for dividing expressions with exponents?
5. How would this process change if you had negative exponents in both the numerator and denominator?

Tip: When dealing with exponents in division, always subtract the exponents of like bases.