What is the difference between radical and exponent?

Question

Is radical a type of exponent? What do we call the power when it is a complex number?

Answer 1

A radical $x^\frac{1}{n}$ (the $n$th root of $x$) is a subset of exponents $x^y$ where $n\in\mathbb{Z}$ and $y\in\mathbb{R}$.

You can still use the terms "exponent" and "power" when $x\in\mathbb{C}$, but radicals are more ill-defined.

Answer 2

They are different things even if related. The sign in front of the radical symbol of power or exponent can be either positive or negative in example given:

$$ e ^ {\pm \sqrt{ x^2- a x + b \sin \omega t} } $$

The radical is a special symbol expressing exponents with different symbolization used for powers and roots only.