Question

if cube root of x + cube root of y + cube root of x is equal to 0 show that x3+ y3 + z
3 -27 XYZ is equal to zero​

Answer 1

Answer:

#given (x⅓+y⅓+z⅓)=0

cubing on both sides

#(x⅓+y⅓+z⅓)³=0³

(a+b+c)³=a³+b³+c³+3(a+b)(b+c)(c+a)

using this formula

#(x⅓+y⅓+z⅓)³=0

#x+y+z+3(x⅓+y⅓)(y⅓+z⅓)(z⅓+x⅓)=0

#x+y+z+3x⅔y⅓+3y⅔z⅓+3z⅔x⅓+3x⅓y⅓z⅓=0

#x+y+z+3x⅓y⅓z⅓(x²+y²+z²)+3x⅓y⅓z⅓=0

#x+y+z+3x⅓y⅓z⅓(x²+y²+z²+1)=0

#x+y+z=-3x⅓y⅓z⅓(x²+y²+z²+1)

#again cubing on both sides,then we get

#x³+y³+z³+3x²y+3y²z+3z²x+3xyz=-27xyz(x²+y²+z²+1)³

solve and bring right side equation to left,then we get

x³+y³+z³-27xyz=0

####note:be careful with plus and minus and with (a+b+c)³ formula