Simplify.

To simplify \(\sqrt{54x^7}\), we start by breaking down the expression under the square root into its prime factors and separating the perfect squares from the non-perfect squares.

1. **Factor 54:**
\[
54 = 2 \times 27 = 2 \times 3^3
\]

2. **Rewrite the expression:**
\[
54x^7 = 2 \times 3^3 \times x^7
\]

3. **Identify perfect squares:**
- From \(3^3\), we can extract \(3^2\) as a perfect square:
\[
3^3 = 3^2 \times 3^1
\]
- For \(x^7\), we can extract \(x^6\) as a perfect square:
\[
x^7 = x^6 \times x^1
\]

4. **Put it all together inside the square root:**
\[
\sqrt{54x^7} = \sqrt{(2 \times 3^2 \times x^6) \times (3^1 \times x^1)} = \sqrt{(2 \times 9 \times x^6) \times (3 \times x)}
\]

5. **Use the property of square roots to separate perfect squares from non-perfect squares:**
\[
\sqrt{54x^7} = \sqrt{2} \cdot \sqrt{9} \cdot \sqrt{x^6} \cdot \sqrt{3} \cdot \sqrt{x}
\]

6. **Calculate the square roots of the perfect squares:**
- \(\sqrt{9} = 3\)
- \(\sqrt{x^6} = x^3\)

7. **Combine the simplified parts:**
\[
\sqrt{54x^7} = 3x^3\sqrt{2 \times 3 \times x} = 3x^3\sqrt{6x}
\]

Thus, the simplified form is:
\[
\sqrt{54x^7} = 3x^3\sqrt{6x}
\]