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## The Power of a Complex Number

If is a complex number and is a natural number, then we can take the power of complex number is a complex number and is a natural number, then we can take the power of complex number is a natural number, then we can take the power of complex number

## Properties of Complex Power

• The magnitude is
• The direction is . Note the argument of the results must be reduced mod . Note the argument of the results must be reduced mod
• De Moivre formula: for n as positive or negative integer. Some authors will make brief function , such that De Moivre formula becomes . for n as positive or negative integer. Some authors will make brief function , such that De Moivre formula becomes . , such that De Moivre formula becomes .
• Parallelogram equality

More general case, if we have two complex numbers and , then we can take the power of complex number and , then we can take the power of complex number , then we can take the power of complex number

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