Answer
$1.48$
Work Step by Step
According to Snell's law,
$$n_1\sin\theta_1=n_2\sin\theta_2$$
where all the angles are measured relative to the normal.
For the red light in quartz,
$$n_{air}\sin\theta_{air}=n_{\rm quartz,red}\sin\theta_{\rm quartz,red}$$
Solving for $\theta_{air}$, to find the incident angle of the white light beam which is common in both colors, red and violet.
$$ \theta_{air}=\sin^{-1}\left[\dfrac{n_{\rm quartz,red}\sin\theta_{\rm quartz,red}}{n_{air}}\right]$$
Plugging the known;
$$\theta_{air}=\sin^{-1}\left[\dfrac{(1.45)\sin26.3^\circ}{1.0}\right]=\bf 40.0^\circ$$
For the violet light in quartz,
$$n_{air}\sin\theta_{air}=n_{\rm quartz,violet }\sin\theta_{\rm quartz,violet }$$
Solving for $n_{\rm quartz,violet }$,
$$n_{\rm quartz,violet }=\dfrac{n_{air}\sin\theta_{air}}{\sin\theta_{\rm quartz,violet }}$$
Plugging the known;
$$n_{\rm quartz,violet }=\dfrac{(1.0)\sin40^\circ}{\sin25.7^\circ}=\color{red}{\bf 1.48}$$
