I have questions about one proof.

It concerns numbers with rational exponents.

Having in mind that for all is then author of book wants to show that must be positive real number.

Proof is:

If is rational number then is also a rational number, so:

Similarly, if is rational number then is rational number, so:

which gives us that . Then is

If then is .

Based on above, we get that which means that must be positive number because is positive number so is positive number.

My question is why must be positive number?

If is rational number then is also a rational number, so:

which means that can be also negative number.

Based on that, then can mean that can be also negative number.