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3 Tutorials that teach Properties of Fractional and Negative Exponents

# Properties of Fractional and Negative Exponents

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Author: Sophia Tutorial
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Simplify an expression with fractional exponents.

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Tutorial
what's covered
1. Properties of Fractional and Negative Exponents

# 1. 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.

formula
Property of Fractional 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.

 Index is denominator Negative exponents from reciprocals

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

 Index is denominator Negative exponent means reciprocals

did you know
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.

 Change to radical, denominator is index, negative means reciprocal Evaluate radical Evaluate exponent Our Solution

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.

big idea

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 rational exponents.

EXAMPLE

 Need common denominator on and Add exponents on and Our Solution

EXAMPLE

 Multiply each exponent by Our Solution

EXAMPLE

 In numerator, need common denominator to add exponents Subtract exponents on x, reduce exponents on y Negative exponent moves down to denominator Our Solution

EXAMPLE

 Need common denominator on x's in parentheses; use Subtract exponents Multiply by Our Solution

summary
It is important to remember that as we simplify with fractional and negative exponents, we are using the 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.

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

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