De Moivre'S Theorem And Nth Roots Worksheet Page 2

Strategy for finding the cube roots of a complex number

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Write the complex number in polar form z = r (cos + i sin )

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Write z in two more equivalent alternative ways by adding 2 to

the argument.

z = r {cos ( + 2 ) + i sin ( + 2 )}

z = r {cos ( + 4 ) + i sin ( + 4 )}

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Write down the cube roots of z by taking the cube root of r and

dividing each of the arguments by 3

NB: the previous strategy gives the three cube roots as

If z = r (cos + i sin ) is written in any further alternative ways such as

z = r {cos ( + 6 ) + i sin ( + 6 )}, this gives a cube root of

which is the same as one of the previously mentioned roots.

It is impossible to find any more.

Example 4

Find the cube roots of 1 + i

First express 1 + i in polar form

and arg (1 + i) =

Hence1 + i can be expressed as

But 1 + i can also be expressed as

Hence, taking the cube root of the modulus and dividing the argument by 3, the cube roots of 1 + i are

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