Let${\mathrm{Z}}_1=(8+\mathrm{i})\sin \mathrm{\theta }+(7+4\mathrm{i})\cos \mathrm{\theta }\text{ and }{\mathrm{Z}}_2=(1+8\mathrm{i})\sin \mathrm{\theta }$ $+(4+7\mathrm{i})\cos \mathrm{\theta }$ are two complex numbers If ${\mathrm{Z}}_1\cdot {\mathrm{Z}}_2=\mathrm{a}+$i b where $\mathrm{a},\mathrm{b}\in \mathrm{R}$ If M is the greatest value of (a + b)$\mathrm{\forall }\mathrm{\theta }\in \mathrm{R}$ then the value of M^1/3 is____________

$\begin{array}{r}{\mathrm{Z}}_1=(8\sin \mathrm{\theta }+7\cos \mathrm{\theta })+\mathrm{i}(\sin \mathrm{\theta }+4\cos \mathrm{\theta })\\ {\mathrm{Z}}_2=(\sin \mathrm{\theta }+4\cos \mathrm{\theta })+\mathrm{i}(8\sin \mathrm{\theta }+7\cos \mathrm{\theta })\end{array}$
Hence,${\mathrm{Z}}_1=\mathrm{x}+\mathrm{iy}\text{ and }{\mathrm{Z}}_2=\mathrm{y}+\mathrm{ix}$
Where $\mathrm{x}=(8\sin \mathrm{\theta }+7\cos \mathrm{\theta })\text{ and }\mathrm{y}=(\sin \mathrm{\theta }+4\cos \mathrm{\theta })$
$\begin{array}{l}{\mathrm{Z}}_1\cdot {\mathrm{Z}}_2=(\mathrm{xy}-\mathrm{xy})+\mathrm{i}\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)=\mathrm{i}\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)=\mathrm{a}+\mathrm{ib}\\ \Rightarrow \mathrm{a}=0;\mathrm{b}={\mathrm{x}}^2+{\mathrm{y}}^2\end{array}$
Now, ${\mathrm{x}}^2+{\mathrm{y}}^2=(8\sin \mathrm{\theta }+7\cos \mathrm{\theta }{)}^2+(\sin \mathrm{\theta }+4\cos \mathrm{\theta }{)}^2$
$\begin{array}{l}=65{\sin }^2\mathrm{\theta }+65{\cos }^2\mathrm{\theta }+120\sin \mathrm{\theta }\cdot \cos \mathrm{\theta }\\ =65+60\sin 2\mathrm{\theta }\\ {\left.\Rightarrow {\mathrm{Z}}_1\cdot {\mathrm{Z}}_2\right|}_{max}=125\end{array}$