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The given expression is:

\[
\frac{6}{\sqrt{5w^6}}
\]

We are asked to express this in simplest radical form with a rational denominator, assuming \( w > 0 \).

### Step-by-Step Solution:

1. **Simplify the denominator**:
   The square root of \( 5w^6 \) can be broken down as:
   \[
   \sqrt{5w^6} = \sqrt{5} \cdot \sqrt{w^6}
   \]

   Since \( \sqrt{w^6} = w^3 \), the denominator becomes:
   \[
   \sqrt{5w^6} = w^3 \cdot \sqrt{5}
   \]

   So, the expression now looks like:
   \[
   \frac{6}{w^3 \cdot \sqrt{5}}
   \]

2. **Rationalize the denominator**:
   To remove the square root from the denominator, we multiply both the numerator and denominator by \( \sqrt{5} \):
   \[
   \frac{6}{w^3 \cdot \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{6\sqrt{5}}{w^3 \cdot 5}
   \]

   Simplifying the denominator:
   \[
   \frac{6\sqrt{5}}{5w^3}
   \]

### Final Answer:
\[
\frac{6\sqrt{5}}{5w^3}
\]

This is the simplest radical form with a rational denominator.

Would you like more details or have any questions about the process?

### Follow-up Questions:
1. What is the general process for rationalizing denominators?
2. How does simplifying radicals with variables differ from simplifying numbers alone?
3. Can you express \(\sqrt{w^6}\) in another form, such as using exponents?
4. How would the solution change if the expression involved a cube root instead of a square root?
5. Why is it important to assume \( w > 0 \) in this case?

### Tip:
When rationalizing a denominator, always multiply by the necessary conjugate or root to eliminate the radical, but remember to apply this operation to both the numerator and denominator.

Express \(\frac{6}{\sqrt{5w^6}}\) in simplest radical form with a rational denominator (assume w > 0).

Learn how to express \(\frac{6}{\sqrt{5w^6}}\) in its simplest radical form with a rational denominator. This problem covers algebraic concepts including radical expressions and rationalizing denominators, suitable for students in Grades 9-12.

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