Lesson Raising a complex number to an integer power

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Raising a complex number to an integer power

Let me remind you that the formula for multiplication of complex numbers in trigonometric form was derived in the lesson Multiplication and Division of complex numbers in the complex plane in this module.

In accordance to this formula,

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E2+=+r%5E2%2A%28cos%282alpha%29%2Bred%28i%29%2Asin%282alpha%29%29

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E2+=+r%5E2%2A%28cos%282alpha%29%2Bred%28i%29%2Asin%282alpha%29%29

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E3+=+r%5E3%2A%28cos%283alpha%29%2Bred%28i%29%2Asin%283alpha%29%29

and generally

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5En+=+r%5En%2A%28cos%28n%2Aalpha%29%2Bred%28i%29%2Asin%28n%2Aalpha%29%29

,

where n is any integer positive number.
This formula is called De Moivre's formula (after Abraham De Moivre, 1667-1754).
The formula is valid for negative integer exponent also, as well as for n=0.
For example,

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E%28-2%29

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E%28-2%29

%281%2F%28r%5E2%29%29%2A%281%2F%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%5E2%29

%281%2F%28r%5E2%29%29%2A%281%2F%28cos%282alpha%29%2Bred%28i%29%2Asin%282alpha%29%29%29

.

But %281%2F%28cos%282alpha%29%2Bred%28i%29%2Asin%282alpha%29%29%29

due to formula for the quotient of two complex numbers 1 and cos%282alpha%29%2Bred%28i%29%2Asin%282alpha%29

, because we can consider 1 as cos%280%29%2Bred%28i%29%2Asin%280%29

.

By combining very first and very last terms in this chain of equalities, you get finally out target statement

%28r%28cos%28alpha%29%2Bred%28i%29%2Asin%28alpha%29%29%29%5E%28-2%29

%28r%5E%28-2%29%2Acos%28-2alpha%29%2Bred%28i%29%2Asin%28-2alpha%29%29%29

.

Similar proof works for n = -1, -3, -4 and so on.

Summary

To raise the complex number to any integral power, raise the modulus to this power and multiply the argument by the exponent of the power.

Examples

1) Calculate the 3-rd power of the complex number z=2*(cos(20°)+i*sin(20°)).

We have z%5E3+=+%282%2A%28cos%28pi%2F9%29%2Bred%28i%29%2Asin%28pi%2F9%29%29%29%5E3

z%5E3+=+%282%2A%28cos%28pi%2F9%29%2Bred%28i%29%2Asin%28pi%2F9%29%29%29%5E3

2%5E3%2A%28cos%28pi%2F3%29%2Bred%28i%29%2Asin%28pi%2F3%29%29

8%2A%281%2F2%2Bred%28i%29%2Asqrt%283%29%2F2%29+=+4%2B4%2Ared%28i%29%2Asqrt%283%29

.

2) Raise to the 10-th power the number z=-1%2F2-red%28i%29%2A%28sqrt%283%29%2F2%29

.
The modulus of the number z is equal to 1; the argument is 240° (regarding the modulus and the argument see the lesson Complex plane in this module).
Hence, the modulus of z%5E10

is equal to 1 and the argument is equal to 2400°= 6*360°+240°, that is the argument is 240°. Thus you have
z%5E10+=+-1%2F2-red%28i%29%2A%28sqrt%283%29%2F2%29

.

For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane
Addition and subtraction of complex numbers in complex plane
Multiplication and division of complex numbers in complex plane
Raising a complex number to an integer power (this lesson)
How to take a root of a complex number
Solution of the quadratic equation with real coefficients on complex domain
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain

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