Saturday, November 18, 2017

Confused with imaginary numbers

In 9th grade I had an argument with my teacher that



${i}^{3}=i$



where $i=\sqrt{-1}$



But my teacher insisted (as is the accepted case) that:




${i}^{3}=-i$



My Solution:



${i}^3=(\sqrt{-1})^3$



${i}^3=\sqrt{(-1)^3}$



${i}^3=\sqrt{-1\times-1\times-1}$




${i}^3=\sqrt{-1}$



${i}^3=i$



Generally accepted solution:



${i}^3=(\sqrt{-1})^3$



${i}^3=\sqrt{-1}\times\sqrt{-1}\times\sqrt{-1}$




${i}^3=-\sqrt{-1}$



${i}^3=-i$



What is so wrong with my approach? Is it not logical?



I am using the positive square root. There seems to be something about the order in which the power should be raised? There must be a logical reason, and I need help understanding it.

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