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yttire: I don't see this as a math problem, but rather an order of precedence problem. So "the answer" is to know the correct order of precedence as the notation demands.

Well, no. It is not a precedence problem, because with real numbers the following identity should hold:

(x^a)^b) = x^(ab) = x^(ba) = (x^b)^a

However, if x < 0 then some of the steps may yield complex numbers, and then powers and roots (and logarithms, etc) are operations that can have multiple answers. None of those answers are wrong, so it is not a precedence issue. In fact, if you were to define a "required" precedence, then you would be eliminating valid answers.

Euler ran into this problem, and apparently (according to Wikipedia) wrestled with it for years. His ultimate solution was to advise that beginning students of algebra learn complex numbers in polar form from the outset, and he wrote a textbook that did just that.

http://en.wikipedia.org/wiki/Complex_number#History

A simple but drastic alternative is to teach that elementary algebra cannot properly solve problems in which a root is taken of a negative number. IIRC, that was the way I was first taught. Now, many decades later, I suspect that that approach might have been a mistake.

It's a difficult pedagogical decision, because some students in the "normal" range of intelligence are just not intellectually equipped to handle anything beyond the simplest forms of algebra. Complex numbers in polar form may be a step too far for them.

Loren

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