# Exponent

An exponent is a number that tells us how many times the base it is attached to is used as a factor. Exponentiation is a mathematical operation in which the base is raised to an exponent.

In 5^{3}, 5 is the base and 3 is the exponent. The terms "exponent" and "power" are often used interchangeably in this context, and we can read 5^{3} as "five to the power of three," which works out to:

5^{3} = 5 × 5 × 5 = 125

In the case of a negative exponent, we can rearrange the expression by taking the reciprocal, then raising it to a positive exponent, as long as the base is a non-zero real number.

If an exponent is not an integer, and is positive, we take the n^{th} root of the base, where n is the exponent.

## Exponent properties

Any non-zero base raised to the power of 0 is 1.

Example

123456789^{0} = 1

0^{0} is a special case that does not have a definite answer; it is most commonly said to equal 1, or is said to be undefined.

Any real number can be the coefficient to an "invisible" variable to the power of 0. In addition, any term to the power of 1 is itself, so we leave off the power because it is not necessary.

Example

x + 5 = x^{1} + 5x^{0}

0 taken to any power greater than 0 is 0. 0 taken to a negative exponent is undefined.

Example

0^{123456789} = 0

Any number taken to the power of 1 equals itself. 1 taken to the power of any number equals 1.

Example

n^{1} = n

1^{n} = 1

Positive numbers taken to any power will always stay positive. Negative numbers raised to an even power will become positive. Negative numbers raised to an odd number will stay negative.

Example

(-4)^{4} = (-4) × (-4) × (-4) × (-4) = 256

-4^{4} = -(4)^{4} = -(4 × 4 × 4 × 4) = -256

(-5)^{3} = (-5) × (-5) × (-5) = -125

Because of order of operations, it is important to check where the negative sign is in an exponent. If the negative sign is outside of the parentheses, the exponent will be negative unless there is an additional negative sign to offset it.