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) 
z^{2} 
= 
r^{2}(cos 2θ + i sin
2θ) 
z^{3} 
= 
r^{3}(cos 3θ + i sin
3θ) 



This pattern leads to the following important theorem, which is named after the
French mathematician Abraham DeMoivre (1667â€“1754).
Theorem
DeMoivreâ€™s Theorem
If z = r(cos θ + i sin θ) is a complex number and n is a positive integer, then
z^{n} = [r(cos θ + i sin θ)]^{n}
= r^{n}(cos nθ + i sin nθ)
Example 1
Finding Powers of a Complex Number
Use DeMoivreâ€™s Theorem to find
.
Solution
First convert to polar form.
Then, by DeMoivreâ€™s Theorem, you have
NOTE
Notice in Example 1 that the
answer is a real number.
