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$\frac{9}{5-\sqrt7}+\frac{22}{7+\sqrt5}-\frac{1}{\sqrt7+\sqrt5}=\\\\=\frac{9\, (5+\sqrt7)}{(5-\sqrt7)(5+\sqrt7)}+\frac{22\, (7-\sqrt5)}{(7+\sqrt5)(7-\sqrt5)}-\frac{\sqrt7-\sqrt5}{(\sqrt7+\sqrt5)(\sqrt7-\sqrt5)}=\\\\=\frac{9(5+\sqrt7)}{25-7}+\frac{22(7-\sqrt5)}{49-5}-\frac{\sqrt7-\sqrt5}{7-5}=\frac{5+\sqrt7}{2}+\frac{7-\sqrt5}{2}-\frac{\sqrt7-\sqrt5}{2}=\\\\=\frac{5+\sqrt7+7-\sqrt5-\sqrt7+\sqrt5}{2}=\frac{5+7}{2}=\frac{12}{2}=6$