Prove the following identities :
`|{:(a,a^(2),a^(3)),(b,b^(2),b^(3)),(c,c^(2),c^(3)):}|=abc(a-b)(b-c)(c-a)`.

If a,b, and c are all different and if |{:(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3)):}| =0 Prove that abc =-1.

If a,b, and c are all different and if |{:(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3)):}| =0 Prove that abc =-1.

Without expanding, prove the following |(a^3+1,a^2,a),(b^3+1,b^2,b),(c^3+1,c^2,c)|=-(a-b)(b-c)(c-a)(abc+1)

Prove that: {:|(a^2,a,b+c),(b^2,b,c+a),(c^2,c,ab)| = -(a+b+c)(a-b)(b-c)(c-a)

Prove that |{:((b+c)^(2), a^(2), bc),((c+a)^(2), b^(2), ca),((a+b)^(2), c^(2), ab):}|= (a-b) (b-c)(c-a)(a + b+c) (a^(2) + b^(2) + c^(2)) .

Without expanding, prove the following |(1,a,a^3),(1,b,b^3),(1,c,c^3)|=(a-b)(b-c)(c-a)(a+b+c)

Prove that: |{:(-2a, a+b, a+c), (b+a,-2b,b+c),(c+a, c+b,-2c):}|=4(a+b)(b+c)(c+a)

Prove that |[[a,b,c],[a^2,b^2,c^2],[a^3,b^3,c^3]]|= abc (a-b)(b-c)(c-a)

Prove that : {:|(a^2a^2-(b-c)^2,bc),(b^2,b^2-(c-a)^2,ca),(c^2,c^2-(a-b)^2,ab)| = (b-c)(c-a)(a-b)(a+b+c)(a^2+b^2+c^2)

Without expanding, prove the following |(0,ab^2,ac^2),(a^2b,0,bc^2),(a^2b,b^2c,0)|=2a^3b^3c^3