# Fractional exponents

While positive integer exponents tell us how many times to multiply the base, and negative exponents tell us how many times to divide by the base, fractional exponents involve a combination of powers and roots. When a base is raised to a fractional exponent, the numerator indicates the power the base is raised to, and the denominator indicates the root the base is raised to. This is expressed as

where b is the base, n indicates the power, and m indicates the root of the fractional exponent.

## General rules and properties

Fractional exponents follow the same rules as other types of exponents. Refer to the exponent rules page to review exponent rules if necessary, as knowing exponent rules can simplify computation of fractional exponents in many cases.

### nth roots

If the numerator of a fractional exponent is 1, the expression is computed as the nth root of the base. For example, a base raised to the power of 1/2 is equivalent to taking the square root of b; when raised to the power of 1/3, it means to take the cubed root of the base, and so on, such that the denominator of the fractional exponent determines which root of the base to compute.

Example

### Power rule

One of the power rules of exponents states that raising a base that is already raised to an exponent, to another exponent, is the same as raising that same base to the product of the exponents:

Using the commutative property, this rule can also be rearranged as follows:

In the context of fractional exponents, this means that the order in which the root or power is computed does not matter. In either case, the result will be the same since a fractional exponent, n/m, can be broken up as: b^{n × 1/m}, and rearranged such that either the power or root is computed first, as per the rule above.

Example

or

Notice that since 8 is a perfect cube, the first computation is significantly simpler to perform, and can reasonably be done without a calculator, as long as we recognize 8 as a perfect cube. While 4096 is also a perfect cube, it may be more difficult for most of us to recognize it as such. Thus, it is important to pay attention to the order in which we perform the operations given a fractional exponent, since sometimes it may be easier to compute the root first, while other times it may be easier to compute the power first.

### Multiplying fractional exponents

In the context of exponents, only expressions with the same base or the same exponent can be simplified using exponent rules. The rules for fractional exponents are the same as those for other types of exponents:

Same base: | |

Same exponent: | |

Examples

1.

2.

### Dividing fractional exponents

The rules around dividing exponents are similar to those for multiplication. Only expressions with the same base or exponent can be simplified using exponent rules. The rules are the same for both fractional exponents as well as other types of exponents:

Same base: | |

Same exponent: | |

Examples

1.

2.

There are other rules and properties regarding fractional exponents. The above are just a few more commonly used ones.