Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Advanced Algebra II: Activities and Homework » Rules of Exponents

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Busbee's Math Materials display tagshide tags

    This collection is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Rules of Exponents

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

Summary: This module provides practice problems designed to develop concepts related to the rules of exponents.

Exercise 1

Here are the first six powers of two.

  • 21=2 2 1 2
  • 22=4 2 2 4
  • 23=8 2 3 8
  • 24=16 2 4 16
  • 25=32 2 5 32
  • 26=64 2 6 64
  • a. If I asked you for 27 2 7 (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term?
  • b. IWrite an algebraic generalization to represent this rule.

Exercise 2

Suppose I want to multiply 25 2 5 times 23 2 3 . Well, 25 2 5 means 2×2×2×2×2, and 23 2 3 means 2×2×2. So we can write the whole thing out like this.

An image showing the expanded meaning of an exponent

  • a. This shows that ( 2523=2[ ] 2 5 2 3 2 [ ]
  • b. Using a similar drawing, demonstrate what 103104 10 3 10 4 must be.
  • c. Now, write an algebraic generalization for this rule.
  • d. Show how your answer to 1b (the “getting from one power of two, to the next in line”) is a special case of the more general rule you came up with in 2c (“multiplying two exponents”).

Exercise 3

Now we turn our attention to division. What is 312310312310 size 12{ { {3 rSup { size 8{"12"} } } over {3 rSup { size 8{"10"} } } } } {}?

  • a. Write it out explicitly. (Like earlier I wrote out explicitly what 2523 2 5 2 3 was: expand the exponents into a big long fraction.)
  • b. Now, cancel all the like terms on the top and the bottom. (That is, divide the top and bottom by all the 3s they have in common.)
  • c. What you are left with is the answer. So fill this in: 312310312310 size 12{ { {3 rSup { size 8{"12"} } } over {3 rSup { size 8{"10"} } } } } {} 3[ ] 3 [ ] .
  • d. Write a generalization that represents this rule.
  • e. Suppose we turn it upside-down. Now, we end up with some 3s on the bottom. Write it out explicitly and cancel 3s, as you did before: 310312310312 size 12{ { {3 rSup { size 8{"10"} } } over {3 rSup { size 8{"12"} } } } } {}= ___________________________ = 1313 size 12{ { {1} over {3 rSup { size 8{ left [~ right ]} } } } } {}
  • f. Write a generalization for the rule in part (e). Be sure to mention when that generalization applies, as opposed to the one in part (d)!

Exercise 4

Use all those generalizations to simplify x3y3x7x5y5x3y3x7x5y5 size 12{ { {x rSup { size 8{3} } y rSup { size 8{3} } x rSup { size 8{7} } } over {x rSup { size 8{5} } y rSup { size 8{5} } } } } {}

Exercise 5

Now we’re going to raise exponents, to exponents. What is 234 2 3 4 ? Well, 23 2 3 means 2×2×2. And when you raise anything to the fourth power, you multiply it by itself, four times. So we’ll multiply that by itself four times:

234 2 3 4 = (2×2×2) (2×2×2) (2×2×2) (2×2×2)

  • a. So, just counting 2s, 234=2[ ] 2 3 4 2 [ ] .
  • b. Expand out 1053 10 5 3 in a similar way, and show what power of 10 it equals.
  • c. Find the algebraic generalization that represents this rule.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks