Question 1067437: Hello . I am a high school student and we just learned about radical and radical notation.
Our teacher says index of radical must be integer and greater than 2 by definition. But I can’t understand why we can’t have radical with negative or rational numbers? For example why we can’t have this? http://imgur.com/a/qWZqG
Our teacher says it’s because negative and rational indexes are not defined for radical notations but why they are not defined? They certainly has answers. any help is greatly appreciated
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! I would answer by convention the index is the number of roots taken, and conventionally we don't have a way of multiplying a number by itself 1.5 times or -1 times.
Another concern is when we get into logs, log2 (4)=2; we don't have negative bases (yes, in complex math they can occur, but in the math that nearly everybody else does, they cannot be.) While -2^2=4 we can't say log (-2)4=2; they aren't defined for negative numbers.
While the 2 means raising the power to (1/2) and the 3 to (1/3) making it a (-1/2) doesn't raise it to the -2 power.
Some things are simply defined, like 0!, which is defined as 1. All the factorials I have ever looked at have never been negative, but there does exist a -1!, but you are very much in the realm of very complex mathematics where few of us ever go.
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