Let A 1 , A 2 , A 3 , … , A 8 be the vertices of a regular octagon that lie on a circle of radius 2 .…

Given,

$A_1,A_2,A_3,\dots ,A_8$ vertices of a regular octagon lying on a circle of radius $2$.

Now using the concept of $n^{th}$ root of unity,

Let any point $P$ be, $Z=\left(2\right)(1{)}^{1/8}$

$\Rightarrow Z^8=2^8\times 1$

$\Rightarrow Z^8-2^8=0$

$\Rightarrow Z=2,2\alpha ,2{\alpha }^2,2{\alpha }^3,\dots ,2{\alpha }^7;\left\{\text{here }\alpha =e^{i\frac{2\pi }{8}}\right\}$

$\Rightarrow Z^8-2^8=(Z-2)(Z-2\alpha )\left(Z-2{\alpha }^2\right)\left(Z-2{\alpha }^3\right)\dots \left(Z-2{\alpha }^7\right)$

$\Rightarrow \left|Z^8-2^8\right|=|Z-2||Z-2\alpha |\dots .\left|Z-2{\alpha }^7\right|$

$\text{ But }\left|Z^8+\left(-2^8\right)\right|\le |Z{|}^8+2^8$

$\begin{array}{r}\Rightarrow |Z-2||Z-2\alpha |\dots \left|Z-2{\alpha }^7\right|\le |Z{|}^8+2^8\\ \le 2^8+2^8 \\ \le 2^9 \end{array}$

$\Rightarrow \mathrm{Max}\left(P{A}_1\cdot P{A}_2\dots .P{A}_8\right)=2^9=512$