|[0,b^(2)a,c^(2)a],[a^(2)b,0,c^(2)b],[a^(2)c,b^(2)c,0]|=2a^(3)b^(3)c^(3)

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

Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3

Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3

If [[a,a^(2),a^(3)-1b,b^(2),b^(3)-1c,c^(2),c^(3)-1]]=0

Prove that ((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3))/((a-b)^(3)+(b-c)^(3)+(c-a)^(3))=(a+b)(b+c)(c+a)

Evaluate |[0,c,b] , [c,0,a] , [b,a,0]| hence show that |[0,c,b] , [c,0,a] , [b,a,0]|^2= |[b^2+c^2,ab,ac] , [ab,c^2+a^2,bc] , [ca,bc,a^2+b^2]|=4a^2b^2c^2

If a,b,c are different and |(a,a^(2),a^(3)-1),(b,b^(2),b^(3)-1),(c,c^(2),c^(3)-1)|=0 then

Suppose a,b,c, gt0 and |(a^(3)-1,a^(2),a),(b^(3)-1,b^(2),b),(c^(3)-1,c^(2),c)|=0 then least possible value of a+b+c is ____________