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# Exponent

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

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

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

It can then be shown that:

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

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

## When p is not a positive integer

When p is a negative integer, $\displaystyle{ a^p }$ means that we are notating the number $\displaystyle{ \large \frac{1}{a*a*a*a...} }$ 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, $\displaystyle{ a^p }$ always equals one.
When p equals -1, $\displaystyle{ a^p }$ 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.