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I enter x^* to typeset x*, but get an error message. Why?
The problem is that * is a binary operator, that is there has to be an expression on both sides of the symbol.
You either have to enter the expression x^{{}*{}} to avoid the error message, or you could type x^"*" as well, where the quotes will interprete the asterisk not as a binary operator but as a "standalone" symbol.
If you use this construction a lot you might want to define the symbol as a userdefined symbol. See FAQ #012. If you have defined the * as a userdefined symbol, say %ast, you can enter x^%ast.
This answer applies to a lot of other symbols as well. Some operators, such as +, are unary, that is there only has to be an expression to the right of the symbol. To typeset x+ you can enter x^{+{}} or, of course, x^"+".