jmc

Squaring the given expression, we have $$\begin{array}{rl}{\left(\sqrt{5+\sqrt{21}}+\sqrt{5-\sqrt{21}}\right)}^2& =(5+\sqrt{21})+(5-\sqrt{21})+2\sqrt{(5+\sqrt{21})(5-\sqrt{21})}\\ & =10+2\sqrt{4}\\ & =14.\end{array}$$Because the given expression is clearly positive, its value must be $\boxed{\sqrt{14}}.$