Question

Use this formula to find all the solutions of the equation? `x^3` + 8 =0

Answer 1

See below.

Explanation:

Using de Moivre's identity

`e^(i phi) = cosphi + i sin phi`

`x^3= -8 = -2^3=2^3e^(i(pi+2kpi))` for `k=0,1,2` then

`x = 2 e^(i(pi+2kpi)/3)` so the roots are:

`2 e^(ipi/3), 2 e^(ipi),2 e^(i(5pi)/3)` or

`1 + i sqrt[3], -2, 1 - i sqrt[3]`