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The given mathematical expression is:

\[
\log_6 \left( \sqrt[3]{6\sqrt{36}} \right)
\]

Let's simplify this step by step:

### Step 1: Simplify \(\sqrt{36}\)
\[
\sqrt{36} = 6
\]

Thus, the expression inside the cube root becomes:
\[
6\sqrt{36} = 6 \cdot 6 = 36
\]

### Step 2: Simplify \(\sqrt[3]{36}\)
The expression becomes:
\[
\log_6 \left( \sqrt[3]{36} \right)
\]

### Step 3: Use logarithmic properties
We can write the logarithm as:
\[
\log_6 \left( \sqrt[3]{36} \right) = \log_6 \left( 36^{1/3} \right)
\]

Using the power property of logarithms:
\[
\log_b (a^c) = c \cdot \log_b (a)
\]

This becomes:
\[
\frac{1}{3} \cdot \log_6 (36)
\]

### Step 4: Simplify \(\log_6 (36)\)
Since \(36 = 6^2\), we have:
\[
\log_6 (36) = \log_6 (6^2) = 2
\]

### Step 5: Final result
Substitute \(\log_6 (36) = 2\) into the expression:
\[
\frac{1}{3} \cdot 2 = \frac{2}{3}
\]

Thus, the value of the expression is:
\[
\boxed{\frac{2}{3}}
\]

Would you like more details or assistance? 

Here are 5 related questions to expand your understanding:
1. How do logarithmic properties like \(\log_b (a^c) = c \cdot \log_b (a)\) work in different bases?
2. Can you explain the general relationship between roots and exponents?
3. How do we handle logarithms with arbitrary bases using change of base formula?
4. What happens when the argument of a logarithm is less than the base, e.g., \(\log_6(5)\)?
5. How can cube roots and other fractional exponents be simplified in nested operations?

**Tip:** Always simplify inner terms first when dealing with nested roots and powers to make calculations more manageable.

Learn how to simplify the expression log_6(∛(6√36)) using properties of logarithms and exponents. Step-by-step solution includes root simplification, logarithmic properties, and exponent rules. Suitable for Grades 10-12 students.

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