# Exponent

An exponent is a number that tells us how many times the base it is attached to is used as a factor.

Example

In 5^{3}, 5 is the base and 3 is the exponent. This works out to:

5^{3} = 5 × 5 × 5 = 125

Exponentiation is a mathematical operation in which the base is raised to an exponent, also known as a power. The exponent is commonly written as a superscript, as shown above.

### exponent properties

Any nonzero base raised to the power of 0 is 1. 0^{0} is a special case that does not have a definite answer.

Example

123456789^{0} = 1

Because of this, 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.

See also negative exponents, order of operations.