4. Find the indicated roots of these complex numbers.

Question

a. 3√1 = 1?
b. sqrt{- 3 - 4i}
c. 4sqrt{81cis60°}
d. 5sqrt{32cis (pi/6)}

Answer 1

I will do two of them, you try the rest
Using De Moivre's Theorem

b.
let z = -3 - 4i
magnitude = √((-3)^2 + (-4)^2) = 5
tanØ = 180 + 53.13 or 233.13° , since -3-4i is in the third quadrant of the
Argand plane
so we can say: z = 5cis233.13°
then z^(1/2) = (-3-4i)^(1/2) = √5cis 233.13/2° = √5 cis 116.565°
There will be 2 roots, each 360/2 or 180° apart, so the roots are
√5cis 116.565° and √5cis296.565
or -1 + 2i and 1 - 2i

let's check the last one algebraically:
(1-2i)^2 = 1 - 4i + 4i^2
= 1 - 4i - 4
= -3 - 4i , just for fun check the other one .

c.

I will assume you want the fourth root of 81cis60°

let z = 81cis60°
then z^(1/4) = 81^4 cis (60/4)°
= 3cis15°
But there will be 4 roots, each 360/4° or 90° apart from each other
so add multiples of 90 to the above answer to get:
3cis15° , 3cis105°, 3cis195, and 3cis285°
(note that adding another 90° would bring us back to 3cis15°)

I don't understand your notation in a.