Math

4. Find the indicated roots of these complex numbers.
a. 3√1 = 1?
b. sqrt{- 3 - 4i}
c. 4sqrt{81cis60°}
d. 5sqrt{32cis (pi/6)}

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  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.

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