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Try ratsimp with algebraic : true

(C1) z : 2/sqrt(2);

(D1) 2/SQRT(2)

(C2) ratsimp(z);

(D2) 2/SQRT(2)

(C3) ratsimp(z),algebraic;

(D3) SQRT(2)

(C4) z : 2/2^(2/3);

(D4) 2/2^(2/3)

(C5) ratsimp(z);

(D5) 2/2^(2/3)

(C6) ratsimp(z),algebraic;

(D6) 2^(1/3)

(C7)