$解：(1)\sqrt{12+2\sqrt{35}}$
$ \begin{aligned} &= \sqrt{(\sqrt{7}+\sqrt5)^2} \\ &=\sqrt7+\sqrt5 \\ \end{aligned}$
$(2)\sqrt{16-4\sqrt{15}}$
$ \begin{aligned} &=\sqrt{2}× \sqrt{8-2\sqrt{15}} \\ &=\sqrt{2}(\sqrt{5}-\sqrt{3}) \\ &= \sqrt{10}-\sqrt{6} \\ \end{aligned}$
$(3)∵a+6\sqrt{5}$
$ \begin{aligned} &=(m+\sqrt{5}n)^2 \\ &=m^2+5n^2+2\sqrt{5}mn \\ \end{aligned}$
$∴a=m^2+5n^2，6=2mn $
$又∵a，m，n为正整数$
$∴m=1，n=3或m=3，n=1$
$∴当m=1，n=3时，a=46$
$当m=3，n=1时，a=14$
$综上所述，a的值为46或14。$
