Exponent

Also referred to as the power a base number is raised to, the exponent is the superscript value of a number written as $a^p$.

Suppose that a is a real number. When the product a × a × a × a is written as $a^4$, the number 4 is the index, or exponent.

When the exponent is a positive integer p, then $a^p$ means a × a × a ... × a where there are p occurrences of a.

It can then be shown that:

(i) $a^p * a^q = a^{p+q}$
(ii) $\frac{a^p}{a^q} = a^{p-q}$, if a is not equal to 0
(iii) $(a^p)^q = a^{pq}$
(iv) $(ab)^p = a^pb^p$

where in (ii) it is required that $p \gt q$.

When p is not a positive integer

When p is a negative integer, $a^p$ means that we are notating the number $\large \frac{1}{a*a*a*a...}$ where, you guess it, the absolute value of p represents the number of occurences of a.
When p equals zero and a does not equal zero, $a^p$ always equals one.
When p equals -1, $a^p$ equals the reciprocal (or the multiplicative inverse) of a, that means 1/a.
When 0 is taken to a negative power, the result will be always undefined, as that implies in division by zero.

00 is sometimes considered undefined, but is normally sensibly defined as 1.