To simplify \( 3\sqrt{54} \), we start by breaking down the square root.
First, we can factor 54 into its prime factors: \[ 54 = 2 \times 3^3 \]
Now, we can rewrite \( \sqrt{54} \): \[ \sqrt{54} = \sqrt{2 \times 3^3} = \sqrt{2} \times \sqrt{3^3} \]
Since \( \sqrt{3^3} = \sqrt{3^2 \times 3} = 3\sqrt{3} \), we can substitute this back: \[ \sqrt{54} = \sqrt{2} \times 3\sqrt{3} = 3\sqrt{6} \]
Now, we go back to our expression \( 3\sqrt{54} \): \[ 3\sqrt{54} = 3(3\sqrt{6}) = 9\sqrt{6} \]
Thus, the simplified form of \( 3\sqrt{54} \) is: \[ \boxed{9\sqrt{6}} \]