$1.55 \sqrt{6 m+2 \sqrt{9 m^{2}-n^{2}}}-\sqrt{6 m-2 \sqrt{9 m^{2}-n^{2}}}=2 \sqrt{3 m-n}$.

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### 1.55 We have

$$
\begin{aligned}
& 6 m + 2 \sqrt{9 m^{2} - n^{2}} = 3 m + n + 2 \sqrt{9 m^{2} - n^{2}} + 3 m - n = \\
& = (\sqrt{3 m + n} + \sqrt{3 m - n})^{2}
\end{aligned}
$$

Similarly,

$$
6 m - 2 \sqrt{9 m^{2} - n^{2}} = (\sqrt{3 m + n} - \sqrt{3 m - n})^{2}
$$

Therefore,

$$
\begin{aligned}
& \sqrt{(\sqrt{3 m + n} + \sqrt{3 m - n})^{2}} - \sqrt{(\sqrt{3 m + n} - \sqrt{3 m - n})^{2}} = \\
& = \sqrt{3 m + n} + \sqrt{3 m - n} - \sqrt{3 m + n} + \sqrt{3 m - n} = \\
& = 2 \sqrt{3 m - n}, \text{ where } m > \frac{n}{3}, m > 0
\end{aligned}
$$