Prove the following identities \[ \left|\begin{array}{ccc} z & x & y \\ z^{2} & x^{2} & y^{2} \\...

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Prove the following identities
\[
\left|\begin{array}{ccc}
z & x & y \\
z^{2} & x^{2} & y^{2} \\
z^{4} & x^{4} & y^{4}
\end{array}\right|=\left|\begin{array}{ccc}
x & y & z \\
x^{2} & y^{2} & z^{2} \\
x^{4} & y^{4} & z^{4}
\end{array}\right|=\left|\begin{array}{ccc}
x^{2} & y^{2} & z^{2} \\
x^{4} & y^{4} & z^{4} \\
x & y & z
\end{array}\right|=x y z(x-y)(y-z)(z-x)(x+y+z) .
\]
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