Prove the identities: `|[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]|` =x y z (x-y)(y-z)(z-x)(x+y+z)

Explanation

To prove the identities

\[ |[z, x, y],[z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| = xyz (x-y)(y-z)(z-x)(x+y+z), \]

we will denote the determinants as \( I_1, I_2, \) and \( I_3 \) respectively.

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