1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Challenge 23: Fractional exponents

  1. Nov 17, 2014 #1
    With only only paper & pencil (no calculator or logarithmic tables), figure out which of the following expressions has a greater value: 101/10 or 31/3.

    Please make use of the spoiler tag and write out your full explanation, not just the answer.
     
    Last edited: Nov 17, 2014
  2. jcsd
  3. Nov 17, 2014 #2
    Set ##x = 10^{1/10}##
    Set ##y=3^{1/3}##

    ##x^{30} = 10^3 = 1000##
    ##y^{30} = 3^{10} = 9^5 > 81^2 > 6400##

    ##3^{1/3} > 10^{1/10}##
     
  4. Nov 17, 2014 #3
    I see that was still too easy :)
     
  5. Nov 17, 2014 #4
    Also...
    for ##y > x > e##, ##x^y > y^x##, since ##\frac{\ln x}{x}## reaches maximum at ##e##.
     
  6. Nov 17, 2014 #5

    ZetaOfThree

    User Avatar
    Gold Member

    Make a guess that ##3^{1/3}>10^{1/10}##. This is true iff ##3^{10} > 10^3 = 1000##. We can see that this is true since ##3^{10} = 27^{3} \cdot 3>27^3 > 10^3##. Therefore we made the right guess that ##3^{1/3}>10^{1/10}##.
     
    Last edited: Nov 17, 2014
  7. Nov 26, 2014 #6

    Curious3141

    User Avatar
    Homework Helper

    The plodding, rigorous way.

    First sketch the curve ##y = x^{\frac{1}{x}}## for real, positive ##x##.

    It can be shown that the function is always positive, starting from the origin, reaching a maximum of ##e^{\frac{1}{e}}## at ##x=e## then decreasing asymptotically to ##1## as ##x \to \infty##. All this can be shown by implicit differentiation and L' Hopital's Rule. There are no other turning points.

    Since ##3## and ##10## are both greater than ##e## and the function is decreasing over this interval, that allows us to conclude that ##3^{\frac{1}{3}} > 10^{\frac{1}{10}}##.

    Taking the ##30##th (which is the lcm of ##3## and ##10##) power is the quick and elementary way, but this is more general.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Challenge 23: Fractional exponents
  1. Fractional exponents (Replies: 4)

Loading...