Question

proves that 2x 3 + 2y 3 + 2z 3 -6 xyz = (x+y+z) {(x-y) 2 + (y-z) 2 + (z-x) 2 } hence evalute : 2(13) 3 + 2(14) 3 + 2(15) 3 - 6 x 13 x 14 x 15

Answer 1

we know according to algebric formula,

a^3+b^3+c^3-3abc=1/2 (a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}
use this here ,
x^3+y^3+z^3-3xyz=1/2(x+y+z){(x-y)^2+(y-z)^2+(z-x)^2}
multiply 2 both side ,
2x^3+2y^3+2z^3-6xyz=(x+y+z){(x-y)^2+(y-z)^2+(z-x)^2}

now ,
2 (13)^3+2 (14)^3+2 (15)^3-6 (13)(14)(15)=
(13+14+15){(13-14)^2+(14-15)^2+(15-13)^2}
=(42){1+1+4)=42 x 6=252