What Should One Teach in Elementary Mathematics?

The course I am working with the students here in Chilpancingo is Advanced Algebra. This is a first year algebra class. Before solving cubics and quartics at the end of the class in the middle of June, we are studying fractionary roots of complex numbers.

Some of them have trouble visualizing complex numbers and their roots. They tried to calculate:

(-1 + i)^-2/5 and (1 -i)^-2/5

I want to tell them about Euler´s formula, e^iθ = cosθ + isinθ, and how to use it for this calculation.

I am reading a book by Lakoff and Núñez which analyzes the mathematical concepts behind Euler´s formula, in particular, What does it mean e^iπ + 1 = 0?

The students used their scientific calculators and a formula they got from previous class notes. Now I want to emphasize this other method.

Euler´s formula has the number e; they don´t know this number, but more importantly, I think, they don´t know the concepts. That is my dilemma. Teach them more powerful concepts, or stick to the calculational methods they already know?

The conflict is that there is at least one online complex number calculator:

Complex Number Calculator.

I feel like emphasizing concepts, and leaving calculational technique to the internet.