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Show that x = 2


Date: 08/29/2001 at 23:03:24
From: Andrew Tretten
Subject: No Calculator Numeric Manipulation

I have the following equation:

   x = (10+(108)^(1/2))^(1/3) + (10-(108)^(1/2))^(1/3) 

I am to show that x = 2 without using a calculator.

Basically, I've tried to approach it the way I would if the roots 
were square roots instead of cubic roots. That is, move one of the 
terms to the "x" side of the equation and cube it to remove the cube 
root from the other side of the equation. The problem is that there 
would still be 2 terms in the binomial expansion that are cube roots 
so I haven't accomplished anything.

Then I tried just cubing the equation to start with, but of course I 
ran into the same problem.  

Lastly I tried dividing by one of the cube root terms to get 

x/(10+(108)^(1/2))^(1/3) = 
              1 + (10-(108)^(1/2))^(1/3)/(10+(108)^(1/2))^(1/3)

That can simplify to 

x/(10+(108)^(1/2))^(1/3) = 1 + (15(3)^(1/2)-26)^(1/3)

Which is certainly a little neater, but it still doesn't get me 
anywhere. Any hints in the right direction?  

I think something was mentioned about fractions like

1/(1+1/(+...)) or something like that, but I'm not sure where that 
would go or really how to use it.

Normally I'm used to symbolic math and when I do get numbers I use 
matlab or a calculator to do calculations.  I'm not sure what to do 
here.

Thanks for any help you can give.
Andy


Date: 08/30/2001 at 08:41:58
From: Doctor Peterson
Subject: Re: No Calculator Numeric Manupilation

Hi, Andy.

The first thing I see here is that the form of the answer reminds me 
of the formula for the roots of a cubic equation. That turns out to be 
relevant, but doesn't help directly in finding the answer.

The second thing I see is that the expressions inside the cube roots 
are very similar. It will help if we call the two cube roots a and b, 
in order to see more clearly what is happening:

    x = a + b

    a^3 = 10 + sqrt(108)

    b^3 = 10 - sqrt(108)

Now I notice that the sum (or difference) of a^3 and b^3 has a simple 
form. I know that a+b and a^3+b^3 are related:

    a^3 + b^3 = (a + b)(a^2 - ab + b^2)

I also notice that ab is a very simple number. See if you can use 
this, plus the fact that

    (a + b)^2 = a^2 + 2ab + b^2

to find a simple equation for which x can be shown to be the only 
solution.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Exponents
High School Polynomials

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