Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Advanced Algebra II: Conceptual Explanations » Properties of Radicals

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.
  • Bookshare

    This collection is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech Initiative

    Comments:

    "DAISY and BRF versions of this collection are available."

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

  • Featured Content display tagshide tags

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

    Comments:

    "This is the "concepts" 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 […]"

    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.
 

Properties of Radicals

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

Summary: This module covers some properties of radicals.

What is x2+9x2+9? Many students will answer quickly that the answer is (x+3)(x+3) and have a very difficult time believing this answer is wrong. But it is wrong.

x2x2 is x*x* 1 and 99 is 3, but x2+9x2+9 is not (x+3)(x+3).

Why not? Remember that x2+9x2+9 is asking a question: “what squared gives the answer x2+9x2+9 ?” So (x+3)(x+3) is not an answer, because (x+3)2 =x2+6x+9(x+3)2=x2+6x+9, not x2+9x2+9 .

As an example, suppose x=4x=4. So x2+9 = 42+9 = 25 =5x2+9=42+9=25=5. But (x+3)=7(x+3)=7.

Common Error:

If two numbers are added or subtracted under a square root, you cannot split them up. In symbols: a+ba+ba+ba+b or, to put it another way, x2 + y2 a+b x2 + y2 a+b

x2+9x2+9 cannot, in fact, be simplified at all. It is a perfectly valid function, but cannot be rewritten in a simpler form.

How about 9x29x2 ? By analogy to the previous discussion, you might expect that this cannot be simplified either. But in fact, it can be simplified:

9x2 =3x9x2=3x

Why? Again, 9x29x2 is asking “what squared gives the answer 9x2 9x2 ?” The answer is 3x3x because (3x)2=9x2 (3x)2=9x2.

Similarly, 9x2 =3x9x2=3x, because 3x2=9x23x2=9x2 size 12{ left ( { {3} over {x} } right ) rSup { size 8{2} } = { {9} over {x rSup { size 8{2} } } } } {}.

On the other hand...:

If two numbers are multiplied or divided under a square root, you can split them up. In symbols: ab = a bab = ab, ab = abab= ab

Footnotes

  1. I’m fudging a bit here: x2x2 is xx only if you ignore negative numbers. For instance, if x=–3x=–3, then x2=9x2=9, and x2x2 is 3; so in that case, x2x2 is not xx. In general, x2 =|x|x2=|x|. However, this subtlety is not relevant to the overall point, which is that you cannot break up two terms that are added under a radical.

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