Question

Is my calculator (as well as other sources) wrong?

I entered `(-1)^(4/3)` into my calculator, it gave me "error". Google gave me an imaginary number. But `(-1)^(4/3)` = `((-1)^4)^(1/3)` = `(1)^(1/3)` = `1`. Wolfram alpha gave me 1 when I did the "real-value root" as opposed to the "principle root" so I assume it has something to do with that. Why (and how) did my calculator mess up?

Answer 1

Your calculator isn't smart enough!

Explanation:

I'm going to assume that your calculator/Google didn't treat the exponent as `(-1^4)^(1/3)` or vice versa, and instead treated it as `-1^(1.bar(333))`. When approached as `-1^(1.bar(333))`, you get an imaginary number in your solution.

I checked this in MATLAB and got a similar experience.

EDIT: It looks like it depends on the order of operations.

In MATLAB, if I plug in:

`((-1)^(1/3))^4=-0.5-0.866i`

However:

`((-1)^4)^(1/3)=1`

So it looks like the difference is how the order of operations is approached. If the cubic root is taken first, you get an imaginary result. If you take the integer exponent first, you get a real number.