Question

Determine which regions contain cube roots of −1. Check all that apply.

Answer 1

cube roots of -1 lies in the region given in the best options are  A. on real axis, B. quadrant 1, F. quadrant 4.

What is cube roots of -1?

Cube roots of -1 mean "finding solution of the equation z³ = -1 since the equation is of order 3, the equation has 3 roots in complex field."

According to the question,

Cube roots of -1 can written as $(-1)^\frac{1}{3}$

First, write the given number in polar form

$x = ( cos 0 - i sin 0)^\frac{1}{3}$

Add 2kπ to the argument

$x = (cos2k\pi - i sin 2k\pi )^\frac{1}{3}$

Apply De Moivre's  theorem

$(cos\alpha - i sin\alpha) = cosn\alpha + i sin\alpha$

x = $\frac{2k\pi }{3} - i sin\frac{2k\pi }{3}$

Put k = 0,1,2..., (n-1)                   [sin (-θ) = -sin θ, cos(-θ) = cos θ]

The three roots are

cos 0 -  sin 0, cos $\frac{2\pi }{3}$, $i sin\frac{2\pi }{3}$ , cos $\frac{4\pi }{3}$,  $i$sin$\frac{4\pi }{3}$

-1, $-\frac{1}{2} ,- i \frac{\sqrt{3} }{2}, -\frac{1}{2} , +\frac{\sqrt{3} }{2}$ ,  are the three roots of cube roots of -1

It can be seen that the three roots lies in the regions of real axis, quadrant 1 and quadrant 4.

Hence, cube roots of -1 lies in the region given in the best options are  A. on real axis, B. quadrant 1, F. quadrant 4.

Learn more about cube roots here:

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