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 > 2t. 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.

BC
Answered by Benjamin C. MAT tutor

1524 Views

See similar MAT University tutors

Related MAT University answers

All answers ▸

Can you please help with Question 5 on the 2008 MAT?


The sequence xn is given by the formula x_n = n^3 − 9n^2 + 631. What is the largest value of n for which x_n > x_(n+1)?


Let a and b be positive real numbers. If x^2 + y^2<=1 then what is the largest that ax+by can get?


The inequality x^4 < 8x^2 + 9 is satisfied precisely when...


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning