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Old 2004-04-27, 13:55   #19
jinydu
 
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Dec 2003
Hopefully Near M48

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Ok, I'll try to apply that technique to the expression:

cbrt(-54+sqrt(2700))+cbrt(-54-sqrt(2700))

a = cbrt(-54+sqrt(2700))
b = cbrt(-54-sqrt(2700))
a^3 + b^3 = -108
a*b = cbrt((-54+sqrt(2700))*(-54-sqrt(2700))) = cbrt(216) = 6
x (the number we're trying to determine) = a + b

x^3 = (a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3 = a^3 + b^3 + 3ab(a+b) = a^3 + b^3 + 3abx

x^3 = -108 + 18x

x^3 - 18x + 108 = 0 (which is exactly the cubic where I got my expression from). This is basically just the reverse of the depressed cubic formula.
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