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^ (power)

Keyword Abbreviation Token (hex) Version(s) Classification
^ none AE 1.0 to 7.0 Operator (1)

base exponent
Parameters Type Legal Value(s) Default Value Note(s)
base Float
any if exponent is an integer
Must be non-negative if exponent if not an integer
exponent Float any if base is non-zero
Must be non-negative is base is zero
Calculate an arithmetic expression.

The power operator typically appears as an up arrow (↑) not a caret (^) in the font of most CBM computers. 
If the exponent is zero, CBM BASIC will always return a value of 1.  This is questionable when the base is zero, but otherwise completely reasonable.  Otherwise (exponent not zero) if the base is zero then CBM BASIC will always return a value of 0.  This is fine when exponent is positive, but completely wrong when exponent is negative!
Otherwise (neither base nor exponent are zero), if the base is negative, then in reality a non-integer exponent will often generate a complex (or pure imaginary) number.  In CBM BASIC, the result is always assumed to be non-real and an ILLEGAL QUANTITY ERROR is generated.
Otherwise, it operates how you might think... at least for integer exponents: just multiply base by itself X number of times, where X is the absolute value of exponent.  If exponent is negative, divide the result into 1.  For fraction exponents, the result is too hard to describe... it general the result is an irrational number (think of the square of 2, if that means anything to you).
Anyway, if the result is greater than 1.70141183e+38 then an OVERLOW ERROR occurs.  If the result is less than 2.93873588e-39 then it silently underflows to zero.
The power operator has the highest fixed priority.  The user of course may over-ride the built-in operator priorities by using parentheses "(" and ")".
Like all operators, the result may be printed, stored in an appropriate variable, or combined with other operators in a larger expression.  In this case, a numeric variable is required for storage.  But if the variable is an integer, the minimum result that can be stored is -32768 and the maximum is +32767; otherwise an ILLEGAL QUANTITY ERROR is generated.
I almost forgot!  You can calculate the nth root of a number x with the expression x ^ (1/n).  Because the exponent (1/n) will not be an integer in the usual case, the x must not be negative (refer to the rules above).
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