解:?$(1)$?由數(shù)軸知?$2 < a < 3$?
?$b=\vert a - \sqrt {10}\vert + \vert 2 - a\vert =\sqrt {10}-a + a - 2=\sqrt {10}-2$?
?$(2)b + 2=\sqrt {10},$??$m=\sqrt {10}-3,$?
?$8 - b=10 - \sqrt {10},$??$n=4 - \sqrt {10}$?
∴?$2\ \mathrm {m} + 2n + 1=3,$?平方根為?$\pm \sqrt 3$?