Answer to Question #224096 in Algebra for Rashmay

Question

How can De Moivre's theorem be described? What is the scope of this theorem? Give two examples for roots and two examples for powers.

Answer 1

How can De Moivre's theorem be described?

De Moivre’s theorem is a way of calculating compound numbers that are only in polar form.

It states that, “for a positive integer $n$ , $Z^n$ is found by raising the modulus to the nth power and

multiplying the argument by n.”

If $Z=r[cos(\theta)+isin(\theta)]$

Then $Z^n=r[cos(n\theta)+isin(n\theta)]$ Where $Z$ is a Complex number.

What is the scope of this theorem?

De Moivre’s theorem applies when finding the roots and powers of complex numbers that are in

polar form. If they are not in polar form, it does not applies.

Give two examples for roots

Given $Z^3=i$ ,then the roots are ${\sqrt 3 \over 2}+i{1\over 2}$ , $-{\sqrt 3 \over 2}+i{1\over 2}$ and $-i$

$\therefore$ The two examples of roots are ${\sqrt 3 \over 2}+i{1\over 2}$ , $-{\sqrt 3 \over 2}+i{1\over 2}$

Two examples for powers.

$i^9=i$ and $i^4=1$