Rational Exponents
Before we rewrite an exponential expression as a radical, we must make
sure that the rational exponent is reduced to lowest terms.
Example 1
Simplify and write using only positive exponents: w1/3
· w(-3/2) = w1/6
Solution
Each factor has the same base, w.
Therefore, add the exponents and keep w
as the base.
Write each fraction with the LCD, 6.
Add the exponents.
Simplify. |
w1/3
· w(-3/2) = w1/6
= w1/3 + (-3/2) + 1/6
= w2/6 - 9/6 + 1/6
= w -6/6
= w -1 |
Use
to write the expression using
a positive exponent. |
![](./articles_imgs/928/ration72.gif) |
Therefore,
![](./articles_imgs/928/ration73.gif)
Example 2
Simplify and write using only positive exponents:
![](./articles_imgs/928/ration74.gif)
Solution
There is more than one way to start simplifying.
We begin with the Power of a Quotient Property. |
![](./articles_imgs/928/ration75.gif) |
Use the Power of a Product Property to raise
each factor to the power -3. |
![](./articles_imgs/928/ration76.gif) |
Use the Power of a Power Property. |
![](./articles_imgs/928/ration77.gif) |
Use the following to write the coefficients with
positive exponents:
![](./articles_imgs/928/ration78.gif) |
![](./articles_imgs/928/ration79.gif) |
Evaluate the coefficients and simplify
![](./articles_imgs/928/ration80.gif) |
![](./articles_imgs/928/ration81.gif) |
So,
![](./articles_imgs/928/ration82.gif)
Note:
We could use radical notation to write
![](./articles_imgs/928/ration83.gif) |