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Fractional Exponents

Module by: Kenny M. Felder. E-mail the author

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

Exercise 1

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, 73757375 size 12{ { {7 rSup { size 8{3} } } over {7 rSup { size 8{5} } } } } {} 7[ ] 7 [ ] .
  • b. But if you write it out and cancel the excess 7s, then 73757375 size 12{ { {7 rSup { size 8{3} } } over {7 rSup { size 8{5} } } } } {}= ——.
  • c. Therefore, since 73757375 size 12{ { {7 rSup { size 8{3} } } over {7 rSup { size 8{5} } } } } {} can only be one thing, we conclude that these two things must be equal: write that equation!

Exercise 2

Now, we’re going to approach fractional exponents the same way. Based on our rules of exponents, 9122 9 1 2 2 =

Exercise 3

So, what does that tell us about 912 9 1 2 ? Well, it is some number that when you square it, you get _______ (*same answer you gave for number 2). So therefore, 912 9 1 2 itself must be:

Exercise 4

Using the same logic, what is 1612 16 1 2 ?

Exercise 5

What is 2512 25 1 2 ?

Exercise 6

What is x12 x 1 2 ?

Exercise 7

Construct a similar argument to show that 812=2 8 1 2 =2.

Exercise 8

What is 2713 27 1 3 ?

Exercise 9

What is −113 1 1 3 ?

Exercise 10

What is x13 x 1 3 ?

Exercise 11

What would you expect x15 x 1 5 to be?

Exercise 12

What is 25−12 25 1 2 ? (You have to combine the rules for negative and fractional exponents here!)

Exercise 13

Check your answer to #12 on your calculator. Did it come out the way you expected?

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

Exercise 14

Using the rules of exponents, 8132=8[ ] 8 1 3 2 8 [ ] .

So that gives us a rule! We know what 8122 8 1 2 2 is, so now we know what 8⅔ is.

Exercise 15

823= 8 2 3 =

Exercise 16

Construct a similar argument to show what 1634 16 3 4 should be.

Exercise 17

Check 1634 16 3 4 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 912 9 1 2 you would say it means 9 9 so it is 3.)

Exercise 18

8−12 = 8 1 2 =

Exercise 19

8−23= 8 2 3 =

For these problems, just say what it means. (For instance, 312 3 1 2 means 3 3 , end of story.)

Exercise 20

10-4 10 -4

Exercise 21

2−34 2 3 4

Exercise 22

xab x a b

Exercise 23

x-ab x -a b

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