Let $\alpha$ and $\beta $ be the roots of ${x^2} + \sqrt 3 x - 16 = 0$, and $\gamma $ and $\delta $ be the roots of ${x^2} + 3x - 1 = 0$. If ${P_n} = $ ${\alpha ^n} + {\beta ^n}$ and ${Q_n} = {\gamma ^n} + {\hat \sigma ^n}$, then $\frac{{{P_{25}} + \sqrt 3 {P_{24}}}}{{2{P_{23}}}} + \frac{{{Q_{25}} - {Q_{23}}}}{{{Q_{24}}}}$ is equal to