Contents of this Precalculus episode:

The differences between real and complex radicals, The n^{th} root of a real number, The n^{th} root of a complex number, Polar form, Trigonometric form.

The differences between real and complex radicals.

Here comes some magic:

The question is: where is the trick?

The fact is: there is no trick.

For example, a while ago we defined what means. We said .

This is in spite of the fact that there is another number whose square is 4: that number is negative 2.

For complex numbers, the situation is much more entertaining.

For example

Yes, but

furthermore:

So, there are four numbers whose fourth power is 1.

This minor inconvenience prompts us to define radicals differently for complex numbers than we did for real numbers.

The nth root of a real number always meant exactly one number. The nth root of a complex number, on the other hand, means all numbers whose nth power is the original number.

For example

real complex

The nth roots of complex number are complex numbers

where the following holds:

and

Here, r denotes the absolute value of the complex number, which is a real number.

So, this is an ordinary real radical - just like in the old days.

RADICALS

Here is this complex number:

Let's see what happens if we search its 5th root.

First, we need the trigonometric form.

And then we are ready for the radical.

This means five complex numbers.

Case k=5 doesn’t matter. We return to case k=0.

So, that’s it for radicals.

The differences between real and complex radicals, The n^{th} root of a real number, The n^{th} root of a complex number, Polar form, Trigonometric form.

Precalculus episode

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