Quote:
Originally Posted by jinydu
For example, simplify the following:
cbrt(54+sqrt(2700))+cbrt(54sqrt(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))

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(54s(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