$ \large\begin{array}{l}  \sqrt {\frac{{1143}}{{32}} + \frac{{105}}{{32}}\sqrt {65} }  = ? \\  a = \frac{1}{2}\sqrt 2 \sqrt {\frac{{1143}}{{32}} - \sqrt {\left( {\frac{{1143}}{{32}}} \right)^2  -  \left( {\frac{{105}}{{32}}} \right)^2  \cdot 65}}  = \frac{5}{8}\sqrt {15}  \\  b = \frac{{\frac{{105}}{{32}}\sqrt 2 }}{2{\sqrt {\frac{{1143}}{{32}} - \sqrt {\left( {\frac{{1143}}{{32}}} \right)^2  - \left( {\frac{{105}}{{32}}} \right)^2  \cdot 65} }}} = \frac{7}{{40}}\sqrt {15}  \\  c = 65 \\  \sqrt {\frac{{1143}}{{32}} + \frac{{105}}{{32}}\sqrt {65} }  = \frac{5}{8}\sqrt {15}  + \frac{7}{{40}}\sqrt {15}  \cdot \sqrt {65}  = \frac{5}{8}\sqrt {15}  + \frac{7}{8}\sqrt {39}  \\  \end{array}$