Complex Numbers Worksheet - Appendix F, Cengage Page 8
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APPENDIX F
Complex Numbers
Powers and Roots of Complex Numbers
To raise a complex number to a power, consider repeated use of the multiplication
rule.
z
r cos
i sin
2
2
z
r
cos 2
i sin2
3
3
z
r
cos 3
i sin 3
This pattern leads to the following important theorem, which is named after the
French mathematician Abraham DeMoivre (1667–1754).
THEOREM F.1
DeMoivre’s Theorem
If
z
r cos
i sin
is a complex number and n is a positive integer, then
n
n
n
z
r cos
i sin
r
cos n
i sin n .
EXAMPLE 9
Finding Powers of a Complex Number
12
Use DeMoivre’s Theorem to find
1
3i
.
Solution
First convert to polar form.
2
2
1
3i
2 cos
i sin
3
3
Then, by DeMoivre’s Theorem, you have
12
2
2
12
1
3i
2 cos
i sin
3
3
2
2
12
2
cos 12
i sin 12
3
3
4096 cos 8
i sin 8
4096.
NOTE Notice in Example 9 that the
answer is a real number.
Recall that a consequence of the Fundamental Theorem of Algebra is that a poly-
nomial equation of degree n has n solutions in the complex number system. Each
solution is an nth root of the equation. The nth root of a complex number is defined
as follows.
Definition of nth Root of a Complex Number
The complex number
u
a
bi
is an nth root of the complex number z if
n
n
z
u
a
bi
.
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