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102
Powers and Roots of Complex Numbers
In this lesson, we present a theorem without proof then use the theorem to find all the
roots (real or non-real) of a polynomial equation. We start with the definition below.
The complex number w  a  bi is an nth root of the complex
n
number z if  a  bi   z .
Now, we state the theorem.
For any positive integer n, then w is an nth root of the complex
number z  r  cos   i sin   if
1
    2k
w  r n cos 
n
 
for k  0,1, 2,

   2k  
  i sin 

n  


, n 1 .
Using this theorem, we can solve the equation x6  1  0 as below.
x6  1  0
x6  1
x 61
From the above work, we see that solving x6  1  0 requires finding the six sixth-roots of 1. To
apply our theorem, we first note that 1  1 cos 0  i sin 0  . Then, we substitute into the
  0  2k
expression w  1 cos 
  6
we obtain,
1
6

 0  2k
  i sin 

 6
  0  2  0 
w0  1 cos 
6
 
1
6

  for the values of k  0,1, 2,


 0  2  0 
  i sin 
6


, 5 . For k  0,
   0 
 0 
   1 cos    i sin     1 .
   6 
 6 
For k  1,
1
  0  2 1 
w1  16 cos 
6
 
3

 0  2 1       
   1
.
  i sin 
   1 cos    i sin      i
6
2


   3 
 3  2
103
For k  2,
  0  2  2  
1
3
 0  2  2      2 
 2  
.
w2  1 cos 
  i sin 
   1 cos 
  i sin 
    i
6
6
2
2


   3 
 3 
 
1
6
For k  3,
1
  0  2  3
w3  16 cos 
6
 

 0  2  3   
  i sin 
   1 cos    i sin     1 .
6



For k  4,
  0  2  4  
 0  2  4 
w4  1 cos 
  i sin 
6
6


 
1
6
   4 
 4
   1 cos 
  i sin 
   3 
 3
1
3

.
    i
2
2

   5
   1 cos 
   3
3
 1
.
   i
2
 2
Finally, for k  5 ,
  0  2  5 
w5  1 cos 
6
 
1
6

 0  2  5 
  i sin 
6


Hence, the six sixth-roots of 1 are 1, 1 ,

 5
  i sin 

 3
1
3
1
3
1
3
1
3
,  i
,  i
, and  i
.
i
2
2
2
2
2
2
2
2
104
Suggested Homework in Dugopolski
Section 7.5:
#41, #42, #43, #44, #51, #52
Suggested Homework in Ratti and McWaters
Section 7.8:
#53, #59, #65-69 odd
Application Exercise
Powers of i possess an interesting periodic property. Since i 2  1 , we obtain
the following pattern.
i 2  1
i 3  i 2  i  1  i  i
i 4  i 2  i 2  1  1  1
i 5  i 4  i  1 i  i
i 6  i 4  i 2  1  1   1
i 7  i 4  i 3  1  i   i
i 8  i 4  i 4  1 1  1
Thus, every natural number power of i can be expressed as one of the numbers i , 1 ,
i , or 1 . Use this fact and the pattern above to simplify i 228 .
Homework Problems
#1 Find the square roots of i .
#2 Find the cube roots of 4  4 3i .
#3 Find all the solutions to x4  8  8i 3  0 .
#4 Find all the solutions to x3  1  0 .
#5 Use identities to find the exact value of sin  58π  .