Answer:
52.
Explanation:
Using the identity a^3 + b^3 = (a + b)(a^2 - ab + b^3):
z^3 + 1/z^3 = (z + 1/z)(z^2 - z *1/z + 1/z^2)
= (z + 1/z) (z^2 - 1 + 1/z^2)
= (z + 1/z) (14 - 1 )
= 13(z + 1/z).
Now using the identity a^2 + b^2 = (a + b)^2 - 2ab:
z^2 + 1/z^2 = ( z + 1/z)^2 - 2(z * 1/z) = 14 So:
( z + 1/z)^2 - 2 = 14
( z + 1/z)^2 = 16
Taking z + 1/z = 4 we therefore have:
z^3 + 1/z^3 = 13 * 4 = 52.