# 5.4 Fractional exponents

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This module provides sample problems designed to develop concepts related to fractional exponents.

On the homework, we demonstrated the rule of negative exponents by building a table. Now, we’re going to demonstrate it another way—by using the rules of exponents.

• ## A

According to the rules of exponents , $\frac{{7}^{3}}{{7}^{5}}$ $7^{\mathrm{\left[ \right]}}$ .
• ## B

But if you write it out and cancel the excess 7s, then $\frac{{7}^{3}}{{7}^{5}}$ = ——.
• ## C

Therefore, since $\frac{{7}^{3}}{{7}^{5}}$ can only be one thing, we conclude that these two things must be equal: write that equation!

Now, we’re going to approach fractional exponents the same way. Based on our rules of exponents , $9^{\left(\frac{1}{2}\right)}^{2}$ =

So, what does that tell us about $9^{\left(\frac{1}{2}\right)}$ ? Well, it is some number that when you square it, you get _______ (*same answer you gave for number 2). So therefore, $9^{\left(\frac{1}{2}\right)}$ itself must be:

Using the same logic, what is $16^{\left(\frac{1}{2}\right)}$ ?

What is $25^{\left(\frac{1}{2}\right)}$ ?

What is $x^{\left(\frac{1}{2}\right)}$ ?

Construct a similar argument to show that $8^{\left(\frac{1}{2}\right)}=2$ .

What is $27^{\left(\frac{1}{3}\right)}$ ?

What is $-1^{\left(\frac{1}{3}\right)}$ ?

What is $x^{\left(\frac{1}{3}\right)}$ ?

What would you expect $x^{\left(\frac{1}{5}\right)}$ to be?

What is $25^{\left(\frac{-1}{2}\right)}$ ? (You have to combine the rules for negative and fractional exponents here!)

OK, we’ve done negative exponents, and fractional exponents—but always with a 1 in the numerator. What if the numerator is not 1?

Using the rules of exponents, $8^{\left(\frac{1}{3}\right)}^{2}=8^{\mathrm{\left[ \right]}}$ .

So that gives us a rule! We know what $8^{\left(\frac{1}{2}\right)}^{2}$ is, so now we know what 8⅔ is.

$8^{\left(\frac{2}{3}\right)}=$

Construct a similar argument to show what $16^{\left(\frac{3}{4}\right)}$ should be.

Check $16^{\left(\frac{3}{4}\right)}$ on your calculator. Did it come out the way you predicted?

Now let’s combine all our rules! For each of the following, say what it means and then say what actual number it is. (For instance, for $9^{\left(\frac{1}{2}\right)}$ you would say it means $\sqrt[]{9}$ so it is 3.)

$8^{\left(\frac{-1}{2}\right)}=$

$8^{\left(\frac{-2}{3}\right)}=$

For these problems, just say what it means. (For instance, $3^{\left(\frac{1}{2}\right)}$ means $\sqrt[]{3}$ , end of story.)

$10^{-4}$

$2^{\left(\frac{-3}{4}\right)}$

$x^{\left(\frac{a}{b}\right)}$

$x^{\left(\frac{\mathrm{-a}}{b}\right)}$

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