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) $a^p\ast a^q=a^{p+q}$

(ii) $\frac{a^p}{a^q}=a^{p-q}$, if a is not equal to 0

(iii) $(a^p{)}^q=a^{pq}$

(iv) $(ab{)}^p=a^p{b}^p$

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

When p is not a positive integer

When p is a negative integer, $a^p$ means that we are notating the number $\frac{1}{a\ast a\ast a\ast a...}$ 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.

External links

• Wikipedia