A trillion is 10^12. Which of the following is bigger: the three trillionth root of 3 or the two trillionth root of 2? You may assume that if 0 < x < y, then 0 < x^n < y^n for integer values of n greater than or equal to 1.

For this sort of question, the easiest approach is to make an educated guess at the correct answer, and prove that this must be correct by contradiction. As taking the root of something has a large effect on its size, we will make an educated guess that the two trillionth root of 2 is larger. Hence, for a contradiction, we will assume that the three trillionth root of 3 is greater than the two trillionth root of 2, and let t = 10¹² (ie, t is a trillion) for the sake of ease of notation.

3^(1/3t) > 2^(1/2t) > 0 implies that (3^(1/3t))^3t > (2^(1/2t))^3t > 0. This implies that 3 > 2^t. This is quite clearly incorrect - 2 to the trillionth power is blatantly not less than 3. Hence our initial assumption was incorrect, and we have proved that the two trillionth root of 2 is larger than the three trillionth root of 3 by contradiction.