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 indexes?
For example why can’t we have either of these?
$$\sqrt[\frac32]8=8^{\frac1{\left(\frac32\right)}}=8^{\frac23}=\sqrt[3]{8^2}=\sqrt[3]{64}=4$$
$$\sqrt[-2]4=4^{\frac1{-2}}=4^{-\frac12}=\sqrt[2]{4^{-1}}=\sqrt[2]{\frac14}=\frac12$$
Our teacher says it’s because negative and rational indexes are not defined for radical notations but why they are not defined? They certainly have answers.
Best Answer
Nothing prevents you from choosing to define a meaning for something like $\sqrt[-2/3]{x}$.
It's just not something that is usually done, because the only "reasonable" choice of definition would to be to make it mean the same as what we already have the notation $x^{-3/2}$ for -- and since the latter notation is simpler and easier to read, there is no particular demand for also writing it $\sqrt[-2/3]{\cdots}$.
In short, the fact that we usually don't define this is not out of any kind of mathematical necessity, but simply because there doesn't seem to be any need to.