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Exponentiation

The exponentiation of a complex number z = x + i y , always gives a binomial development, as

zⁿ = (x + i y)ⁿ    

 

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Explanation

We start with the binomial development

z⁰ = (x + i y)⁰ = 1
z¹ = (x + i y)¹ = x + i y
z² = (x + i y)² = x² + 2i xy − y²

 

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Example 1

We calculate the complex number (1 + i)¹⁰ by

and because i² = –1 this gives

so that

 

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