Simplify.

To simplify the expression \(\sqrt{108a^6}\), we start by factoring out the perfect square terms from inside the square root.

1. **Factor the number 108:**
\[
108 = 36 \times 3 = 6^2 \times 3
\]

2. **Factor \(a^6\):**
\[
a^6 = (a^3)^2
\]

Now we can rewrite \(\sqrt{108a^6}\) by substituting these factors into the expression:

\[
\sqrt{108a^6} = \sqrt{36 \times 3 \times (a^3)^2}
\]

3. **Apply the property of square roots:**
Using the property \(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\), we can separate the square root into perfect squares and non-perfect squares:

\[
\sqrt{108a^6} = \sqrt{36} \cdot \sqrt{3} \cdot \sqrt{(a^3)^2}
\]

4. **Calculate the square roots of the perfect squares:**
\[
\sqrt{36} = 6 \quad \text{and} \quad \sqrt{(a^3)^2} = a^3
\]

5. **Combine the results:**
\[
\sqrt{108a^6} = 6 \cdot a^3 \cdot \sqrt{3}
\]

6. **Final answer:**
Therefore, the simplified form is:

\[
\sqrt{108a^6} = 6a^3\sqrt{3}
\]