How do you simplify #root3(1)#?
The cubed root of 1 is the same as raising 1 to the power of
Working in the reals we get
Every non-zero complex number has three cube roots, so there
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: