# How do you simplify #root3(1)#?

##### 2 Answers

#### Answer:

#### Explanation:

The cubed root of 1 is the same as raising 1 to the power of

#### Answer:

Working in the reals we get

Every non-zero complex number has three cube roots, so there

#### Explanation:

If we're working in real numbers we just note

One of the odd things we find out when we delve into complex numbers is that the function

The key fact is Euler's Identity squared. I call it **Euler's True Identity.**

Euler's True Identity shows

We can raise Euler's True Identity to any integer power

What's all this got to do with the cube root of one? It's the key. It tells there are a countably infinite number of ways of writing one. Some of them have different cube roots than others. It's why non-integer exponents give rise to multiple values.

That's all a big windup. Usually I just start these by writing:

The last step is of course Euler's Formula

Since we have the

So we get three values for the cube root of one: