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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

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    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

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    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.
     
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