Use Sophia to knock out your gen-ed requirements quickly and affordably. Learn more
×

Properties of Fractional and Negative Exponents

Author: Sophia

what's covered
In this lesson, you will learn how to simplify an expression with fractional exponents. Specifically, this lesson will cover:

Table of Contents

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 to know
Properties of Fractional Exponents
Rule space # 1 colon thin space m-th root of a equals a to the power of 1 over m end exponent
Rule space # 2 colon thin space open parentheses m-th root of a close parentheses to the power of n equals a to the power of n over m end exponent

When converting between radical to exponent, 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.

EXAMPLE

Notice how the index of the radical becomes the denominator of the fraction:

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

EXAMPLE

Notice how the negative exponents come from reciprocals:

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

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

EXAMPLE

Again, note how the denominator of the exponent becomes the index of the radical:

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

EXAMPLE

Again, note how the negative exponent means a reciprocal:

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

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.

EXAMPLE

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


2. Using All Exponent Properties

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 General Form
Product Property a to the power of m a to the power of n equals a to the power of m plus n end exponent
Quotient Property a to the power of m over a to the power of n equals a to the power of m minus n end exponent
Power of a Power Property 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
Power of a Product Property open parentheses a b close parentheses to the power of m equals a to the power of m b to the power of m
Power of a Quotient Property 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
Zero Property of Exponents a to the power of 0 equals 1
Properties of Negative Exponents a to the power of short dash m end exponent equals 1 over a 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 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.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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

Properties of Fractional Exponents

Rule space # 1 colon thin space m-th root of a equals a to the power of bevelled 1 over m end exponent

Rule space # 2 colon thin space open parentheses m-th root of a close parentheses to the power of n equals a to the power of bevelled n over m end exponent

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