Table of Contents |
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.
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:EXAMPLE
Notice how the negative exponents come from reciprocals: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:EXAMPLE
Again, note how the negative exponent means a reciprocal: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
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.
Properties of Exponents | General Form |
---|---|
Product Property | |
Quotient Property | |
Power of a Power Property | |
Power of a Product Property | |
Power of a Quotient Property | |
Zero Property of Exponents | |
Properties of Negative Exponents |
|
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 |
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