If x + y + z = 18, xyz = 81 and xy + yz + zx = 90, then find the value of \(\sqrt[4]{x^3+y^3+z^3+xyz}\)

6

x³ + y³ + z³ - 3xyz = (x + y + z)[(x + y + z)² - 3(xy + yz + za)]

= x³ + y³ + z³ - 3 × 81 = 18[(18)² - 3 × 90]

= x³ + y³ + z³ - 243 = 18[324 - 270]

= x³ + y³ + z³ = 18 × 54 + 243 = 1215

The value of \(\sqrt[4]{x^3+y^3+z^3+xyz}\) = \(\sqrt[4]{1215 + 81}\) = 6