Question

Which expression is a sixth root of -1 + i√3?

A.6rt2(cos(90°) + isin(90°))
B. 6rt2 (cos(60°) + isin(60°))
C. 6rt2 (cos(300°) + sin(300°))
D. 6rt2 (cos(20°) + isin(20°))
Pls help

Answer 1

`-1 + i\sqrt3` lies in the second quadrant of the complex plane, so its argument is

`\arg(-1 + i\sqrt3) = \pi - \tan^(-1)\left(-\sqrt3\right) = \frac{2\pi}3 = 120^\circ`

Then any sixth root of
`-1+i\sqrt3` will have an argument of

`\frac{120^\circ + 360^\circ n}6`

where
`n\in\{0,1,2,3,4,5\}`.

When
`n=0`, we have

`\frac{120^\circ}6 = 20^\circ`

so D is a sixth root.

The other sixth roots are separated by arguments of 60 degrees (80, 140, 200, 260, and 320).