De Moivre'S Theorem And Nth Roots Worksheet

De Moivre's theorem and nth roots

De Moivre's theorem is not only true for the integers but can be extended to fractions.

De Moivre's theorem for fractional powers

Example 1

Calculate

By De Moivre's theorem for fractional powers

Example 2

Calculate

By De Moivre's theorem

Example 3

Using De Moivre's theorem calculate

De Moivre's theorem gives

= 2 3 + 2i

The previous worked example showed that

That is,

is a cube root of

This cube root is obtained by dividing the argument of the original number by 3

However, the cube roots of

are complex numbers z which satisfy z

= 1 and so by the

Fundamental theorem of algebra, since this equation is of degree 3, there should be 3 roots. That is, in general,

a complex number should have 3 cube roots.

Given a complex number these 3 cube roots can always be found

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