Powers of Complex Numbers
To raise a complex number to a power, consider repeated use of the multiplication
||r(cos θ + i sin θ)
||r2(cos 2θ + i sin
||r3(cos 3θ + i sin
This pattern leads to the following important theorem, which is named after the
French mathematician Abraham DeMoivre (1667â€“1754).
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)
Finding Powers of a Complex Number
Use DeMoivreâ€™s Theorem to find
First convert to polar form.
Then, by DeMoivreâ€™s Theorem, you have
Notice in Example 1 that the
answer is a real number.