Powers of Complex Numbers
To raise a complex number to a power, consider repeated use of the multiplication
rule.
| z |
= |
r(cos θ + i sin θ) |
| z2 |
= |
r2(cos 2θ + i sin
2θ) |
| z3 |
= |
r3(cos 3θ + i sin
3θ) |
| |
 |
|
This pattern leads to the following important theorem, which is named after the
French mathematician Abraham DeMoivre (1667–1754).
Theorem
DeMoivre’s Theorem
If z = r(cos θ + i sin θ) is a complex number and n is a positive integer, then
zn = [r(cos θ + i sin θ)]n
= rn(cos nθ + i sin nθ)
Example 1
Finding Powers of a Complex Number
Use DeMoivre’s Theorem to find
.
Solution
First convert to polar form.
Then, by DeMoivre’s Theorem, you have
NOTE
Notice in Example 1 that the
answer is a real number.
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