How to find the square root of (7-24i) i complex number

Question

Square root  of this complex number

Answer 1

Use De Moivre's Theorem

z = 7 - 24i

r = ||7 - 24i||

r = 25

θ ≈ 73.79°

 

z^1/2 ≈ 25^1/2(cos 73.79/2 + i sin 73.79/2)

z^1/2  ≈ 5(cos 36.90 = i sin 36.90)

z^1/2 ≈ 5(0.8 + i 0.6)

z^1/2 ≈ 4 + 3i

 

or

 

z^1/2 ≈ 25^1/2( cos (73.79 + 360)/2 + i sin (73.79 + 360)/2)

 

You can do the calculations

 

Answer 2

Zicon,

 

This is the same as finding the two solutions to the equation z² = 7 - 24i.

 

Here are the steps. When you have read them, I recommend trying them yourself and then ask questions if you are still having trouble with this problem.

 

1) Write 7 -24i in trigonometric form. (This looks like r(cos θ + i sin θ) and is shortened to r cis θ by some people.)

2) Set up the formula for the two square roots in trigonometric form. They have the form

 

w_k = √r cis ((θ + 360°k)/2) for k = 1 and k = 2  (assuming degrees for the angles)

 

3) Plug in k = 1 and k = 2 into this formula and simplify the angle(s).

4) Evaluate and rewrite in standard form (a + bi) if that is required.

 

If you need any help with any of these steps, please send a reply as quickly as you can so you can get the problem done.