Rational exponents in the real number system?

[Stephen Tashi]

Are there rigorous texts that treat the topic of raising real numbers to rational powers without treating it a special case of using complex numbers?

I'm not trying to avoid the complex numbers for my own personal use! My goal is to determine whether students who have not studied complex analysis can find authoritative answers.

In casual discussions, people are quick to offer answers to questions such as:
1) Is $(-8)^{4/6}$ equal to $(-8)^{2/3}$?
2) Is $(1)^{5/0}$ defined as the 5th power of the zeroth root of 1?

but what texts give authoritative answers to such questions and develop definitions for the above notations in a manner that is self-consistent?

Usually when the topic of raising a negative number to a rational power comes up, the answers refer to complex numbers. Has any author taken up the thankless task of treating this topic in a rigorous fashion without using complex analysis?

 

[Vanadium 50]

in a rigorous fashion without using complex analysis?

"Rigor" means different things to different people. What do you think is legit and what do you not? In short - can you describe what an answer would look like?

 

[Stephen Tashi]

"Rigor" means different things to different people. What do you think is legit and what do you not? In short - can you describe what an answer would look like?

I imagine a text on "Real analysis" that defines the exponentiation of negative numbers. As a non- example, I don't find such a definition in "baby Rudin" ( Principles of Mathematical Analysis ).