Question

find the cube root of (-6)​

Answer 1

Step-by-step explanation:

Let us call one of these roots z=x+iy, where i=−1−−−√

If z is a cube root of 6, then z3=(x+iy)3

=x3+3ix2y+3i2xy2+i3y3

=x3+3ix2y−3xy2−iy3

=x(x2−3y2)+iy(3x2−y2)

As z3=6,

x(x2−3y2)=6

y(3x2−y2)=0→y=0 OR y2=3x2

If y=0, then z is the real number x; we already know this is 1.81712

If y2=3x2, then (from 1.), −8x3=6→x=−frac126–√3≈−0.90856

From 2., y=±3–√x≈±1.57367

The cube roots (to 6 decimal plsces) are thus:

1.81712

−0.90856+1.57367i