Online College Courses for Credit

+
3 Tutorials that teach Properties of Fractional and Negative Exponents
Take your pick:
Properties of Fractional and Negative Exponents

Properties of Fractional and Negative Exponents

Author: Sophia Tutorial
Description:

Simplify an expression with fractional exponents.

(more)
See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

Begin Free Trial
No credit card required

37 Sophia partners guarantee credit transfer.

299 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 32 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

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
a to the power of n over m end exponent equals left parenthesis m-th root of a right parenthesis to the power of n

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.

open parentheses fifth root of x close parentheses cubed equals x to the power of 3 over 5 end exponent open parentheses root index 6 of 3 x end root close parentheses to the power of 5 equals open parentheses 3 x close parentheses to the power of 5 over 6 end exponent

Index is denominator
1 over open parentheses root index 7 of a close parentheses cubed equals a to the power of short dash 3 over 7 end exponent 1 over open parentheses cube root of x y end root close parentheses squared equals open parentheses x y close parentheses to the power of short dash 2 over 3 end exponent Negative exponents from reciprocals

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

a to the power of 5 over 3 end exponent equals open parentheses cube root of a close parentheses to the power of 5 open parentheses 2 m n close parentheses to the power of 2 over 7 end exponent equals open parentheses root index 7 of 2 m n end root close parentheses squared

Index is denominator
x to the power of short dash 4 over 5 end exponent equals 1 over open parentheses fifth root of x close parentheses to the power of 4 open parentheses x y close parentheses to the power of short dash 2 over 9 end exponent equals 1 over open parentheses root index 9 of x y end root close parentheses squared 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 1 third times 9 to the power of p to represent 9 to the power of 1 third end exponent. 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.

27 to the power of short dash 4 over 3 end exponent
Change to radical, denominator is index, negative means reciprocal
1 over open parentheses cube root of 27 close parentheses to the power of 4
Evaluate radical
1 over open parentheses 3 close parentheses to the power of 4
Evaluate exponent
1 over 81
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

Properties of Exponents
a to the power of m a to the power of n equals a to the power of m plus n end exponent open parentheses a b close parentheses to the power of m equals a to the power of m b to the power of m a to the power of short dash m end exponent equals 1 over a to the power of m
a to the power of m over a to the power of n equals a to the power of m minus n end exponent open parentheses a over b close parentheses to the power of m equals a to the power of m over b to the power of m 1 over a to the power of short dash m end exponent equals a to the power of m

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent

a to the power of 0 equals 1
open parentheses a over b close parentheses to the power of short dash m end exponent equals b to the power of m over a to the power of m

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

a to the power of 2 over 3 end exponent b to the power of 1 half end exponent a to the power of 1 over 6 end exponent b to the power of 1 fifth end exponent
Need common denominator on a apostrophe s open parentheses 6 close parentheses and b apostrophe s open parentheses 10 close parentheses
a to the power of 4 over 6 end exponent b to the power of 5 over 10 end exponent a to the power of 1 over 6 end exponent b to the power of 2 over 10 end exponent
Add exponents on a apostrophe s and b apostrophe s
a to the power of 5 over 6 end exponent b to the power of 7 over 10 end exponent

Our Solution

EXAMPLE

open parentheses x to the power of 1 third end exponent y to the power of 2 over 5 end exponent close parentheses to the power of 3 over 4 end exponent
Multiply each exponent by 3 over 4
x to the power of 1 fourth end exponent y to the power of 3 over 10 end exponent

Our Solution


EXAMPLE

fraction numerator x squared y to the power of begin display style 2 over 3 end style end exponent times 2 x to the power of begin display style 1 half end style end exponent y to the power of begin display style 5 over 6 end style end exponent over denominator x to the power of begin display style 7 over 2 end style end exponent y to the power of 0 end fraction

In numerator, need common denominator to add exponents
fraction numerator 2 x to the power of begin display style 5 over 2 end style end exponent y to the power of begin display style 9 over 6 end style end exponent over denominator x to the power of begin display style 7 over 2 end style end exponent end fraction

Subtract exponents on x, reduce exponents on y
2 x to the power of short dash 1 end exponent y to the power of 3 over 2 end exponent

Negative exponent moves down to denominator
fraction numerator 2 y to the power of begin display style 3 over 2 end style end exponent over denominator x end fraction

Our Solution

EXAMPLE

open parentheses x to the power of begin display style 3 over 2 end style end exponent over x to the power of begin display style 1 third end style end exponent close parentheses to the power of 1 fourth end exponent


Need common denominator on x's in parentheses; use open parentheses 6 close parentheses
open parentheses x to the power of begin display style 9 over 6 end style end exponent over x to the power of begin display style 2 over 6 end style end exponent close parentheses to the power of 1 fourth end exponent


Subtract exponents
open parentheses x to the power of 7 over 6 end exponent close parentheses to the power of 1 fourth end exponent
Multiply 7 over 6 by 1 fourth
x to the power of 7 over 24 end exponent

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

open parentheses a to the power of n close parentheses to the power of m equals a to the power of n m end exponent

Power of a Product Property of Exponents

open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n

Power of a Quotient Property of Exponents

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

Product Property of Exponents

a to the power of n times a to the power of m equals a to the power of n plus m end exponent

Property of Fractional Exponents

a to the power of n over m end exponent equals m-th root of a to the power of n end root

Quotient Property of Exponents

a to the power of n over a to the power of m equals a to the power of n minus m end exponent