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Question 24
If $$x + y + z = 11, x^2 + y^2 + z^2 = 133$$ and $$x^3 + y^3 + z^3 = 881$$, then the value of $$\sqrt[3]{xyz}$$ is:
Solution
$$(x + y + z)^2 = x^2 +Â y^2 +Â z^2 + 2(xy +Â yz +Â xz)$$
$$(11)^2 = 133 + 2(xy +Â yz + xz)$$
2(xy + yz + xz) = -12
xy + yz + xz = -6
$$x^3 + y^3 + z^3 - 3xyz = (x + y +Â z)(x^2 + y^2 + z^2 - (xy + yz + xz)) $$
881 - 3(xyz) = 11(133 +Â 6)
3xyz = 648
xyz = -216
$$\sqrt[3]{xyz}$$ = -6
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