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

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

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

It can then be shown that:

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

where in (ii) it is required that [math]\displaystyle{ p \gt q }[/math].

When p is not a positive integer

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

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