What's the square root of 1+i\sqrt{3}?

Cheexorgeny

Cheexorgeny

Answered question

2022-01-16

What's the square root of 1+i3?

Answer & Explanation

Rita Miller

Rita Miller

Beginner2022-01-17Added 28 answers

Step 1
Notice that square root is a multi-valued function,
1+i3=2ei(π3+2nπ)
nZ
Therefore,
1+i3=2ei(π6+nπ)
Putting n=0,1 yields two solutions, 32+i12 and 32i12
Elaine Verrett

Elaine Verrett

Beginner2022-01-18Added 41 answers

Step 1
Let's let
z=1+i3
This is a complex number, and there are 2 ways we can write a complex number. The first is:
z=x+iy
which separates its real and imaginary parts, and the second is its polar form:
z=reiθ
The number we have been given is currently in the first form, but let’s put it into its polar form:
r=x2+y2=12+32=1+3=4=2
(note that r is a radius and is strictly positive, hence we pick the positive root)
The angle is given by:
arctanyx=arctan31=π3
So the polar form of the complex number
z=1+i3
is:
z=2eiπ3
Now let’s take the square root:
Recall that :
x12=x
(xy)12=x12y12
(ea)b=eab
So if we apply these rules and compute z12
z12=212(eiπ3)12
z12=±2eiπ6
Now recall the beautiful Euler’s formula:
eiθ=cosθ+isinθ
eiπ6=cosπ6+isinπ6
=32+i12
Therefore
z=±2(32+i12)
=±(62+i22)

alenahelenash

alenahelenash

Expert2022-01-24Added 556 answers

Step 11+i3=21+i32=2ei(π3+2πn)Therefore(1+i3)12=(2ei(π3+2πn))12=2ei(π6+πn)=±2eiπ6=±3+i2This isn't really a square root though, per se.

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