# Exponentiation

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Exponentiation is a mathematical operation involving raising a base to an exponent. The operation is defined for positive integer exponents as repeated multiplication of the base by itself. In particular:

$5^3=\underbrace{5\times5\times5}_{3\text{ times}}=125$
$a^5=\underbrace{a\times a\times a\times a\times a}_{5\text{ times}}$
$3^n=\underbrace{3\times3\times3\cdots\times3\times3}_{n\text{ times}}$
$a^b=\underbrace{a\times a\times a\cdots\times a\times a}_{b\text{ times}}$

In the last example above, $a$ is the base and $b$ the exponent (or, less formally, power).

## Other exponents

### Negative exponents

A negative exponent is defined as one over the base to the power of the exponent. For instance,

$x^{-2}=\frac1{x^2}$

### Fractional exponents

A fractional exponent is the value of the denominator root of the base to the power of the numerator. For example:

\begin{align}x^\frac12&=\sqrt{x}\\x^\frac23&=(\sqrt[3]{x})^2\end{align}

## Common exponents

### Square

"Squaring" a value means raising it to an exponent of 2, which is merely multiplying the value by itself.

$a^2=a\times a$

This term, and this simple case of exponentiation, holds a great deal of practicality and implications throughout basic mathematical concepts. Its commonality is emphasized by the fact that many calculators have a "square" button, but not necessarily the capacity for generalized exponentiation.

The term comes from the geometric figure it is named after. The area of any square is the length of its side squared.

## Cube

"Cubing" a value means raising it to an exponent of 3. Again, the term comes originates from the geometric figure. The volume of a cube is equal to the length of a side cubed.

$a^3=a\times a\times a$