If $x + y + z = 17, xyz = 171$ and $xy + yz + 2zx = 111$,then the value of $\sqrt[3]{(x^{3}+y^{3}+z^{3}+xyz)}$ is:

-4

x + y + z = 17

xy + yz + zx = 111

xyz = 171 

We know that,

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

= 17² = x2 + y2 + z² + 2 × 111

= x² + y² + z²  = 289 – 222

= x² + y² + z²   = 67

We also know that,

x³ + y³ + z³ – 3xyz = (x + y + z)[x2 + y2 + z2​ – (xy + yz + zx)]

= x³ + y³ + z³  – 3 × 171 = 17 × (67 – 111)

= x³ + y³ + z³  – 513 = -748

= x³ + y³ + z³  = -748 + 513

= x³ + y³ + z³  = -235

$\sqrt[3]{(x^{3}+y^{3}+z^{3}+xyz)}$ = $\sqrt[3]{-235 +171)}$ = -4