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Homework: Fractional Exponents

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

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

We have come up with the following definitions.

  • x0=1 x 0 1
  • xa= 1xa x a =1xa size 12{ { {1} over {x rSup { size 8{a} } } } } {}
  • xab =xab x a b =xab size 12{ nroot { size 8{b} } {x rSup { size 8{a} } } } {}

Let’s get a bit of practice using these definitions.

Exercise 1

10012 100 1 2

Exercise 2

1002 100 2

Exercise 3

100-12 100 -1 2

Exercise 4

10032 100 3 2

Exercise 5

100-32 100 -3 2

Exercise 6

Check all of your answers above on your calculator. If any of them did not come out right, figure out what went wrong, and fix it!

Exercise 7

Solve for x x: xx1712 xx size 12{ { {x rSup { size 8{ {3} wideslash {2} } } } over {x rSup { size 8{ {1} wideslash {2} } } } } } {} 17 1 2 1712 17 1 2

Exercise 8

Solve for x x: x12=9 x 1 2 9

Exercise 9

Simplify: xxxx size 12{ { {x} over { sqrt {x} } } } {}

Exercise 10

Simplify: x+xx+1xx+xx+1x size 12{ { {x rSup { size 8{ {3} wideslash {2} } } + sqrt {x} } over {x rSup { size 8{ {5} wideslash {2} } } + { {1} over { sqrt {x} } } } } } {}

Hint:

Multiply the top and bottom by x12 x 1 2 .

Now…remember inverse functions? You find them by switching the x x and the y y and then solving for y y. Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in #9, you will start by writing y=x12 y x 1 2 . Switch the x x and the y y, and you get x=y12 x y 1 2 . Now what? Well, remember what that means: it means x=y x y . Once you’ve done that, you can solve for y y, right?

Exercise 11

x3 x 3

  • a. Find the inverse function.
  • b. Test it.

Exercise 12

x-2 x -2

  • a. Find the inverse function.
  • b. Test it.

Exercise 13

x0 x 0

  • a. Find the inverse function.
  • b. Test it.

Exercise 14

Can you find a generalization about the inverse function of an exponent?

Exercise 15

Graph y=2x y 2 x by plotting points. Make sure to include both positive and negative xx values.

Exercise 16

Graph y=2×2x y 2 2 x by doubling all the y-values in the graph of y=2x y 2 x .

Exercise 17

Graph y=2x+1 y 2 x 1 by taking the graph y=2x y 2 x and “shifting” it to the left by one.

Exercise 18

Graph y=12x y 1 2 x by plotting points. Make sure to include both positive and negative xx values.

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