Example 8 Simplify $\sqrt{11+2(1+5)(1+\sqrt{7})}$
$\begin{array}{l}\text { Solution: The radicand }=1^{2}+(\sqrt{5})^{2}+(\sqrt{7})^{2} \\ \quad+2 \sqrt{5}+2 \sqrt{7}+2 \sqrt{5} \cdot \sqrt{7} \\ \quad=(1+\sqrt{5}+\sqrt{7})^{2} . \\ \therefore \text { the expression }=1+\sqrt{5}+\sqrt{7} .\end{array}$