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 $Q_n = \gamma^n + \delta^n$, then $\dfrac{2^{5+\sqrt{3}}\,P_{24} + 2^{3}\,Q_{25}}{Q_{23} + Q_{24}}$ is equal to: