\[\sqrt{87 - 16\sqrt{23}} - \sqrt{39 - 8\sqrt{23}} =\]
\[= \left| 8 - \sqrt{23} \right| - \left| \sqrt{23} - 4 \right| =\]
\[= 8 - \sqrt{23} - \sqrt{23} + 4 =\]
\[= 12 - 2\sqrt{3}\ \]