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Archive for the ‘Radicals’ Category

How to Divide ANY Number by a Radical — Fast!


 

Here’s a super-quick shortcut for  DIVIDING ANY NUMBER by a RADICAL. 

Note: I’m using this symbol () to mean square root.
So √5 means the square root of 5;  √b means the square root of b.
 And … if you want to learn why this shortcut works, see my explanation at the end of the blog.

This shortcut lets you mentally do problems like the following three problems. That means you can do such problems in your mind rather than having to work them out on paper.

     a)  12 / √3 

     b)  10 / √2

     c)  22 / √5

Here are three terms I’ll use in explaining this shortcut.

In a problem like 12 divided by √3, which I am writing as:  12 / √3,

     12  is  the dividend,

     3  is  the number under the radical,

     √3  is  the radical.

The Shortcut, Used for  12 / √3:

  1.  Divide the dividend by the number under the radical.
    In this case, 12 / 3 = 4. 
  2. Take the answer you got, 4, and multiply it by the radical.
    4 x √3  =  4√3

  3. Shake your head in amazement because that, right there, is the ANSWER!

Another Example:  10 / √2

  1.  Divide the dividend by the number under the radical.
    In this case:   10 / 2  =  5
  2. Take the answer you got, 5, and multiply it by the radical.
    5 x √2  =  5√2.  (Don’t forget to shake head in amazement!)

Third Example:  22 / √5

  1.  Divide the dividend by the number under the radical.
    In this case,  22 divided by 5 = 22/5  (Yup, sometimes you wind up with a fraction or with a decimal; that’s why I’m giving an example like this.)
  2. Take the answer you got, 22/5, and multiply it by the radical.
    22/5 x √5 =  22/5 √5.  [Note: the √5 is in the numerator, not
    in the denominator. To make the location of this √5 clear, it’s best
    to write the answer:  2√5 / 5].


NOW TRY YOUR HAND by doing
these PRACTICE PROBLEMS:

a)   18 / √3  

b)   16 / √2  

c)   30 / √5  

d)   10 / √3  

e)   12 / √5

– – – – – – – – – – – – – – – – – –

ANSWERS:

a)   18 / √3  = 6√3

b)   16 / √2  = 8√2

c)   30 / √5  = 6√5

d)   10 / √3  = 10√3/3

e)   12 / √5  = 12√5/5

– – – – – – – – – – – – – – – – – –

WHY THE SHORTCUT WORKS:

The shortcut works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the shortcut in general terms, with:

     a  =  the dividend,

     b  =  the number under the radical,

     √b  =  the radical.

a / √b

=   a
    √b

=   a     √b    =   a √b
    √b   √b            b

Notice: we started with:  a / √b.

And keeping things equal, we ended up with  a √b / b.

This shows that the shortcut works in general. So it works in all specific cases as well!

– – – – – – – – – – – – – – – – – –

Final note: the number under the radical is called the radicand. But that term is so close to the term radical that I thought it would be less confusing if I just called this the number under the radical.

 

 

 

 

 

 

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“Algebra Survival” Program, v. 2.0, has just arrived!


The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.

 

Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.

 

 

 

 

 

 

 

Invisible Misunderstandings: Square roots of 2 and 3


Would you say that the square root of two is an important number in math? Hmmm … and would you agree that the square root of three, while perhaps not quite so important, is still a quantity whose value students should be able to estimate?

Why not, right? After all, these numbers play key roles in the 30-60-90 and 45-45-90 “special triangles.” And therefore they both appear a lot in geometry, and a great deal in trig. And on top of that, root two, widely believed to be the first irrational number discovered, shows up in a wide range of other math contexts as well.

First letter of a text about the square root o...

square root of 2 w/ "parent" triangle

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