- #1

Saturnine Zero

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## Homework Statement

I'm trying to understand negative bases raised to rational powers, when calculating principle roots for real numbers. I'm not worried about complex solutions numbers at this stage. I just can't find a concise explanation I can understand anywhere. I'm self learning as an adult so I don't have a teacher to ask.

## Homework Equations

When, in general is a negative base raised to a rational power undefined for real numbers?

## The Attempt at a Solution

##(-x)^{\frac {odd}{odd}}## I have this as being a real number but reversing the sign of x

##(-x)^{\frac {even}{odd}}## I have this as being a real number but reversing the sign of x

##(-x)^{\frac {odd}{even}}## I have this as being undefined

##(-x)^{\frac {even}{even}}## I have this as being undefined

But I am still confused. For instance the following example ##(-3)^{\frac 2 4}## I'm not sure how to think about it.

##(-3)^{\frac 2 4}## is this ##(-3^2)^{\frac 1 4}## which would be ##9^{\frac 1 4}## which would have a real root?

Or would it be ##(-3^{\frac 1 4})^{2}## and since you can't take the 4th root of (-3) you can't square it so it is undefined?

##(-3)^{\frac 3 4}## I think I understand as it's either ##(-3^3)^{\frac 1 4}## which is trying to take an even root of an odd number, so undefined. Or it's ##(-3^{\frac 1 4})^{3}## which is trying to take an even root of an odd number and then can't be raised to the 3rd power, so is undefined.

Am I on the right track or am I way off?

edit: fixed the latex