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May 21st, 2009, 02:56 PM  #1 
Member Joined: Feb 2008 Posts: 89 Thanks: 0  simplifying sqrt[a + sqrt(b)]
[color=#000080]Greetings: Is there a means by which to simplify the likes of sqrt[2 + sqrt(3)]. That is, can we rewrite this with a single radical? Thanks, Rich B.[/color] 
May 21st, 2009, 05:50 PM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: simplifying sqrt[a + sqrt(b)]
Generally speaking, no. Sorry! 
May 22nd, 2009, 02:11 AM  #3  
Member Joined: Nov 2007 Posts: 50 Thanks: 0  Re: simplifying sqrt[a + sqrt(b)] Quote:
For instance you can try to look for (or any form which seems plausible) fulfilling and compare coefficients at the roots. The expression can be simplified as follows: Notice (this is the initial guess) that This yields Sometimes you can also use some trigonometric identities. We have Now as see here we get which is the same result as before. (Although, I do not see any quick derivation of the formula for that I copied from wikipedia.)  
June 25th, 2009, 03:18 AM  #4 
Newbie Joined: Jun 2009 Posts: 20 Thanks: 0  Re: simplifying sqrt[a + sqrt(b)]
You can also check whether its minimal polynomial is reducable. is a root of the polynomial If this polynomial is irreducable then is its minimum polynomial and this will not help you to to simplify the expression . However if can be reduced to a product of 2 polynomials of degree 2, say , then is either a root of or . It's then easy to check with the abcformula wich of the 4 roots correspond with . To give an example: is a root of the polynomal Notice that can be reduced to , so is either a root of g or h. By checking all roots of g and h with the abcformula: (roots of g) and (roots of h) we find: . However the sad part, your expression can not be simplified this way Then you have to stick to kompiks method. 

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