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

1482 Views

See similar MAT University tutors

Related MAT University answers

All answers ▸

Deduce a formula (in terms of n) for the following sum: sum (2^i * i) where 1<=i<=n, n,i: natural numbers ( one can write this sum as: 1*2^1+ 2*2^2+ .. +n*2^n)


Show that the inequality x^4 < 8x^2 + 9 is satisfied for when -3 < x < 3 .


How many distinct solutions does the following equation have? log(base x^2 +2) (4-5x^2 - 6x^3) = 2 a)None, b)1, c)2, d)4, e)Infinitely many


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


We're here to help

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

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences