Why are rational and real powers only defined for strictly positive bases?

Answer

Dear Tim,

We must make an important distinction for this question. After all, the answer to this question depends on the glasses through which you look. Let’s look at the edit

a^b

where a is a real number and b is a real number.

A) I look through my real glasses: Let’s suppose that b is a rational number and can therefore be written as p/q. In that case, we can^b write as (a^p)^1/qor in other words, we take the qth root of a^p.

if a^p is a negative number, you can only draw a qth root of power if q is an odd number. After all, in that case you can bring out -1 so that you actually take the qth root of a positive number.
If q is even, you cannot take q-th root of power of negative numbers.

With irrational exponents (= non rational , real exponents) we cannot do the above.
The outcome of the operation is therefore not always defined.

B) you will only get a real answer to this question if you look through your complex glasses. I don’t know what your prior knowledge about complex numbers is, but the key to the answer to your question is there.

Without giving a full introduction to complex numbers here, I would like to briefly explain how you can look at it.

The complex numbers are an extension of the real numbers. In other words, all real numbers are complex numbers, but there are many more complex numbers that are not real. In the places where you got the answer in A) that there is no result, there is a complex number that is the result of the operation, but then a non-real complex number.

In short, any complex number can be written in trigonometric notation by a modulus (= a positive number, also called the radius) and a argument (=an angle), so also the number a from our statement.

The complex value of a^b is then given by taking the modulus of a raised to the power of b and multiplying the argument of a by b.

In the case where a is a positive real number, the modulus of a is equal to a and the angle is equal to 0, so we get a^b get as modulus and 0 as argument. The result is then simply the positive real number a^b. So the complex power is really an extension of the power that is already there.

I must point out, however, that there are still many difficulties involved. For example, the argument of a complex number is not unambiguously determined (after all, it is an angle and eg 20 degrees = 380 degrees) and we can get different answers. The convention is then usually that it is assumed that the argument of a always lies between 0 and 360 degrees.

Moreover, it is also possible to take a power of any complex number (even with a complex exponent).

The part about complex numbers may be hard to follow, but I hope this answer makes you wiser.