Quote:
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Originally Posted by jinydu
For example, simplify the following:
cbrt(-54+sqrt(2700))+cbrt(-54-sqrt(2700))
This time, I will not give out the answer a priori (although this particular case was also handpicked carefully). Also, the technique (maybe algorithm is an even better word) you use should also work for:
cbrt(10 + sqrt(108)) + cbrt(10 - sqrt(108))
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I'll use my previous approach, but a little more streamlined notation. Follow my earlier posting if you need more detail.
This time, a = c(-54+s(2700)) and b = c(-54-s(2700)), where x = a+b again.
Once more x^3 = (a+b)^3 = a^3 + b^3 + 3xab
x^3 = -108 + 3x * c(54^2 - 2700) = -108 +3x c(216) = -108 +18x.
Alternatively, x^3 -18x +108 = 0.
If this cubic has a rational solution, it must be an integer (because it is monic) and it must be a divisor of 108, which is 2*2*2*3*3*3. There are very few of these and we don't need to search them all because c(-54+s(2700)) + c(-54 -s(2700)) is a root. The first of these is about c(-54+52) and the second about c(-54 -52). The first term is about -1, the second about c(-106). We know that 4^3 =64 and 5^3 = 125, so we are looking for a root fairly close to -6. The only candidates from the factorization of 108 are -4, -6 and -8 with the numerical estimate strongly suggesting -6.
The answer, of course, is -6.
Paul