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Properties of Fractional and Negative Exponents

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Sophia Tutorial

This lesson extends exponential properties to exponents which are fractional or negative.

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Properties of Fractional and Negative Exponents

When we simplify radicals with exponents, we divide the exponent by the index. Another way to write division is with a fraction bar. This idea is how we will define rational exponents.

The denominator of a rational exponent becomes the index on our radical. Likewise the index on the radical becomes the denominator of the exponent. We can use this property to change any radical expression into an exponential expression.

We can also change any rational exponent into a radical expression by using the denominator as the index.

Nicole Oresme, a Mathematician born in Normandy was the first to use rational exponents. He used the notation to represent . However his notation went largely unnoticed.

The ability to change between exponential expressions and radical expressions allows us to evaluate problems we had no means of evaluating before by changing to a radical.

The largest advantage of being able to change a radical expression into an exponential expression is we are now allowed to use all our exponent properties to simplify. The following table reviews all of our exponent properties.

When adding and subtracting with fractions we need to be sure to have a common denominator. When multiplying we only need to multiply the numerators together and denominators together. The following examples show several different problems, using different properties to simplify the rational exponents.

It is important to remember that as we simplify with rational exponents we are using the exact same properties we used when simplifying integer exponents. The only difference is we need to follow our rules for fractions as well. It may be worth reviewing your notes on exponent properties to be sure you’re comfortable with using the properties.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html

Formulas to Know

- Power of a Power Property of Exponents
- Power of a Product Property of Exponents
- Power of a Quotient Property of Exponents
- Product Property of Exponents
- Property of Fractional Exponents
- Quotient Property of Exponents