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:

-6

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