SOLUTION: I'm having trouble with the following example problem: Find the square roots of the complex number. (Enter your answers as a comma-separated list.) −10i

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Question 1206982: I'm having trouble with the following example problem:
Find the square roots of the complex number. (Enter your answers as a comma-separated list.)
−10i
Found 4 solutions by MathLover1, ikleyn, Edwin McCravy, mccravyedwin:
Answer by MathLover1(20850) About Me  (Show Source):
Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.
I'm having trouble with the following example problem:
Find the square roots of the complex number. (Enter your answers as a comma-separated list.)
−10i
~~~~~~~~~~~~~~~~~~~~ 

Apply deMoivre formula. The square root of -10i has two different values.

Their modulus is sqrt%2810%29.  Their arguments are  3pi%2F4  and 3pi%2F4%2Bpi = 7pi%2F4.


So, one value of the square root is  sqrt%2810%29%2A%28cos%283pi%2F4%29+%2B+i%2Asin%283pi%2F4%29%29 = sqrt%2810%29%2A%28-sqrt%282%29%2F2+%2B+i%2A%28sqrt%282%29%2F2%29%29%29 = -sqrt%2820%29%2F2+%2B+i%2A%28sqrt%2820%29%2F2%29%29.


The other value of the square root is  sqrt%2810%29%2A%28cos%287pi%2F4%29+%2B+i%2Asin%287pi%2F4%29%29 = sqrt%2810%29%2A%28sqrt%282%29%2F2+-+i%2A%28sqrt%282%29%2F2%29%29%29 = sqrt%2820%29%2F2+-+i%2A%28sqrt%2820%29%2F2%29%29.


ANSWER.  Two values of the square root of -10i are  -sqrt%2820%29%2F2+%2B+i%2A%28sqrt%2820%29%2F2%29%29  and  sqrt%2820%29%2F2+-+i%2A%28sqrt%2820%29%2F2%29%29.

Solved.

---------------

On deMoivre formula see your textbook or the lessons
    - How to take a root of a complex number
    - Solved problems on de'Moivre formula
in this site.

Other relevant lessons in this site, related to square roots of complex numbers are
    - How to take a square root of a complex number
    - Solved problem on taking square root of complex number
Learn the subject from there.


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What @MathLover1 wrote in her post, is irrelevant to the solution of the problem.
For safety of your mind, simply ignore her post.



Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!

Ikleyn should have simplified

 sqrt%2820%29%2F2+-+i%2Aexpr%28sqrt%2820%29%2F2%29%29

 sqrt%284%2A5%29%2F2+-+i%2Aexpr%28sqrt%284%2A5%29%2F2%29%29

 2sqrt%285%29%2F2+-+i%2Aexpr%282sqrt%285%29%2F2%29%29

 

 sqrt%285%29+-+i%2Asqrt%285%29

And similarly, the other solution is

 -sqrt%285%29+%2B+i%2Asqrt%285%29

Edwin

Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!

Here's another way to do this problem:

Assume

a%2Bbi%22%22=%22%22sqrt%28-10i%29, where a and b are real numbers.

a%5E2%2B2a%2Ab%2Ai%2Bb%5E2%2Ai%5E2%22%22=%22%22-10i

a%5E2%2B2a%2Ab%2Ai%2Bb%5E2%2A%28-1%29%22%22=%22%22-10i

a%5E2%2B2a%2Ab%2Ai-b%5E2%22%22=%22%22-10i

Set real parts on the left and right equal, and same with imaginary parts:

system%28a%5E2-b%5E2=0%2C2ab=-10%29

system%28a%5E2=b%5E2%2Cab=-5%29

system%28a=%22%22+%2B-+b%2Cab=-5%29

ab=-5 tells us that a and b have opposite signs, thus a=-b

a%5E2=b%5E2
a=sqrt%285%29 and b=-sqrt%285%29
or
a=-sqrt%285%29 and b=sqrt%285%29

Thus the answers are

sqrt%285%29-i%2Asqrt%285%29 and -sqrt%285%29+%2B+i%2Asqrt%285%29

Edwin