If $x + y + z =13, x^{2} + y^{2} + z^{2} = 133$ and $x^{3} + y^{3} + z^{3} = 847$, then the value of $\sqrt[3]{xyz}$ is:
Answer & explanation
Correct answer: option 4
x + y + z = 13,
x2 + y2 + z2 = 133
x3 + y3 + z3 = 847
x3 + y3 + z3 - 3xyz = (x + y + z) ( x2 + y2 + z2 - (xy + yz + zx) ) ----(A)
(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx) ----(B)
From Equation (B)
= 132 = 133 + 2(xy + yz + zx)
= 169 - 133 = 2(xy + yz+ zx)
= (xy + yz + zx) = 18
Put the values in eq. A
x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - (xy + yz + zx) )
= 847 - 3xyz = 13(133 - 18)
= 847 - 115×13 = 3(xyz)
= - 648 = 3(xyz)
= xyz = -216
$\sqrt[3]{xyz}$ = $\sqrt[3]{-216}$ = -6