# Negative exponents

A negative exponent is equal to the reciprocal of the base of the negative exponent raised to the positive power. This is expressed as

where b is the base, and n is the power. In other words, a negative exponent indicates the inverse operation from a positive integer exponent: it indicates how many times to divide by the base, rather than multiply. This is the equivalent of taking the reciprocal of the base (if the base is b, the reciprocal is b^{-1} = ), removing the negative sign, then computing the positive exponent as you would normally. Refer to the following pages for other exponent cases or rules. Briefly, a positive integer exponent indicates how many times to multiply by the base. For example, given the power 2^{5}, we would multiply 2 five times:

2^{5} = 2 × 2 × 2 × 2 × 2 = 32

In contrast, a negative integer exponent can be computed by multiplying by the reciprocal of the base, n times. For example:

2^{-5} = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32

We can see that this aligns with the formula above since 2^{-5} = 1/2^{5}.

Another way to confirm this is using the property of exponents that states:

We know that b^{-m} = 1/b^{m}, so we can move the b^{m} to the numerator by taking the reciprocal, then adding a negative sign:

Below are a few examples of computing negative exponents given different cases.

### Negative base

Computing a negative exponent with a negative base is very similar, and just requires us to remember the rule that a negative base raised to an even exponent results in an even number, while a negative base raised to an odd exponent results in an odd number.

1. (-2)^{-2}:

2. (-2)^{-3}:

### Fractions

Working with negative exponents in fractions involves taking the reciprocal of the base with the negative exponent. If the base is in the numerator, move it to the denominator and remove the negative sign on the exponent; if the base is in the denominator, move it to the numerator and remove the negative sign on the exponent.

Examples

1. :

2. :

### Did you know??

In other mathematical contexts, it is not uncommon for an exponent of -1 to indicate taking the inverse. For example, the inverse of matrix A, which can by symbolized using [A], is [A]^{-1}. This does not mean that the inverse of [A] is the reciprocal of [A]. The "-1" exponent is just a commonly used notation to indicate an inverse operation.