Simplifying Radicals and Exponents

Date: 9/3/95 at 11:23:20
From: Richard Seguin
Subject: Help on "Developing Operations: Radicals" 

In my math book I have a question like this:
                                                         ____
A 1) Which of the following Radicals are equivalant to \/ 32

IT gives me these choices:

   ___       ____      _____      _____      _____
3\/16  , -2\/ 8   , 4\/ 2    , 2\/ 64   , 3\/ 320

                                                             _____
Could you please explain why and how these can be equal to \/ 32

The other thing I had a problem with was this question:

a) 2^1/4

if this doesn't look right it's supposed to be this

2 TO THE POWER OF 1/4

How do you get that low of a answer? 

Richard Seguin 

Date: 9/4/95 at 16:2:34
From: Doctor Ethan
Subject: Re: Help on 

Hey Richard,
	These are neat questions.  I hope that my explanations make sense 
to you.

1.  To get this we have to understand a little about radicals.

Here are a few rules that you need to know.  I am going to write them 
using variables to show that they are good for any numbers.
  _______      _      _      _      _  
\/a*b*c*d  = \/a  * \/b  * \/c  * \/d 

The next rule follows from it.
  _____      _
\/a^2*b = a\/b
                _____      ___       _          _      _
I got this by \/a^2*b  = \/a^2   * \/b = a *  \/b = a\/b

Now using these rules lets look at problem 1.
  __      __      _          _       _
\/32  = \/16  * \/2  = 4 * \/2  = 4\/2
              __       __
Now you try \/27 and \/18  which work the same way.

For your second question, I guess I have less to say.

I understand 2^1/4 just fine but I can't really tell you much about it.

If you want a decimal approximation, I can give you that.

It is  1.18920711500272106672

Other than that there isn't much to say.

2^1/4 is the number that when taken to the fourth power, equals two.

There really isn't much more.

-Doctor Ethan,  The Geometry Forum

Date: 9/5/95 at 5:47:16
From: Richard Seguin
Subject: Re: Help on 

>Okay now for your second question I guess I have less to say.
>
>I understand 2^1/4 just fine but I can't really tell you much about it.
>
>If you want a decimal approximation , I can give you that.
>
>It is  1.18920711500272106672
>
>Other than that there isn't much to say
>
>2^1/4 is the number that when taken to the fourth power, equals two.
>
>There really isn't much more.

        Well how did you come to this answer. I know what it comes to 
but how did you come across a answer like that? It couldn't have been 
from memory right?

Richard Seguin 

Date: 9/13/95 at 15:15:38
From: Doctor Ethan
Subject: Re: Help on 

You are right it definitely wasn't from memory.

I actually got the answer from a computer.  But here is
another way to think about it that would allow you to use a calulator 
or pencil and paper to get the answer.

2^(1/4) squared is 2^1/2  Do you know why?

Well, 2^1/2 is 1.414028......  
(that is from memory but it is really close to that).

So now to find 2^1/4 we just need to take the square root of 1.414028

Or if we are starting from 2 we can just take the square root twice
and get the fourth root of 2  or 2^1/4

Hope that explains it a little.

Maybe you feel like it is cheating to just say use a calculator to 
take the square root.  If you feel that way, there are ways to 
approximate square roots by hand, but they are not fun so I would 
just get comfortable with a calculator.

-Doctor Ethan,  The Geometry Forum

Date: 9/14/95 at 17:10:48
From: Doctor Ken
Subject: Re: Help on 

Richard -

Let me just add to what Ethan told you.  If you have a number to the 1/n 
power, that's the same as the nth root of that number.  So something 
like 5^(1/7) is the 7th root of 5.  So if you have any number to the p/q 
power, that's the same as the pth power of the qth root of the number: 
thus 8^(2/3) = 4.

Why?  Because if you multiply x^(1/n) times itself n times, you get x.  
Therefore it must be the nth root of x.

- Doctor Ken,  The Geometry Forum