What is the simplified form of the square root of 48n^9

$$\\\sqrt{48n^9}=\sqrt{48}*\sqrt{n^9}\\
\sqrt{48n^9}=\sqrt{48}*\sqrt{n^8*n}\\
\sqrt{48n^9}=\sqrt{48}*\sqrt{n^8}\sqrt{n}\\
\sqrt{48n^9}=\sqrt{16*3}*(n^8)^{1/2}\sqrt{n}\\
\sqrt{48n^9}=\sqrt{16}\sqrt{3}*n^4*\sqrt{n}\\
\sqrt{48n^9}=4n^4\sqrt{3}*\sqrt{n}\\
\sqrt{48n^9}=4n^4\sqrt{3n}\\$$

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