20040326, 11:29  #1 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Does Anyone Know how to Simplify the Following Expression?
x  sqrt(x^2  1)
Thanks 
20040326, 13:25  #2 
Jun 2003
2^{2}×61 Posts 
as x gets very large you get 0 :)

20040326, 14:00  #3 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
I mean, an exact simplification.

20040330, 00:42  #4 
"Richard B. Woods"
Aug 2002
Wisconsin USA
1110000010100_{2} Posts 
This is almost certainly not what you want, and is not exactly a simplification, but "Factoris" at http://wims.unice.fr/~wims/wims.cgi?...ebra/factor.en gives:
The factorization of x(x^{2}1)^{0.5} = (0  i) + x + (0 + 0.5i) x^{2 }+ (0.0 + 0.125i) x^{4 }+ (0.0 + 0.0625i) x^{6 }+ O(x^{8}) ;) Last fiddled with by cheesehead on 20040330 at 00:45 
20040330, 02:41  #5 
"Mark"
Apr 2003
Between here and the
2·2,971 Posts 
It appears to be something out of trigonometry, so I looked it up and found:
arccosh(x) = ln(x  sqrt(x²  1)) so e^arccosh(x) = x  sqrt(x²  1) 
20040330, 09:35  #6 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
Thanks for the help, but rogue's formula doesn't work for x = 2.0996011

20040330, 14:24  #7 
"Mark"
Apr 2003
Between here and the
2×2,971 Posts 
I don't have a calculator that can do arccosh. What do both terms evaluate to?

20040330, 14:42  #8 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
LHS ~= 3.945765984
RHS ~= 0.2534362159 
20040330, 17:40  #9 
"Mark"
Apr 2003
Between here and the
5942_{10} Posts 
Oddly, the website I found this in has '', but others have '+'. Using '+' instead of '', you get
1) arccosh(x) = ln(x + sqrt(x^2  1)) 2) e^arccosh(x) = x + sqrt(x^2  1) 3) x  sqrt(x^2  1) = (x  sqrt(x^2  1)) * (x + sqrt(x^2  1))/(x + sqrt(x^2  1)) 4) x  sqrt(x^2  1) = (x^2  (x^2  1))/(x + sqrt(x^2  1)) 5) x  sqrt(x^2  1) = 1/(x + sqrt(x^2  1)) 6) x  sqrt(x^2  1) = 1/(e^arccosh(x)) 7) x  sqrt(x^2  1) = e^arccosh(x) I suspect that e^arccosh(x) might be simplified as well, but I'll leave that for the next person. Last fiddled with by rogue on 20040330 at 17:42 
20040402, 01:03  #10  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×599 Posts 
Quote:
Using that one, my calculator shows LHS = RHS = 3.94576598405592039632709017264425 Alternatively, you can try the negative arccosh with the positive square root, in which case LHS = RHS = 0.253436215944079603672909827355751 Last fiddled with by cheesehead on 20040402 at 01:11 

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