Results 1 to 14 of 14
Like Tree5Thanks
  • 1 Post By greg1313
  • 1 Post By Cervesa
  • 1 Post By Zexuo
  • 1 Post By Archie
  • 1 Post By greg1313

Thread: Which radical expression is larger?

  1. #1
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Which radical expression is larger?

    $\displaystyle \sqrt[22]{10} \ \ \ or \ \ \ \sqrt[30]{30}$

    The type is relatively small. That is the equivalent of 10^(1/22) versus 30^(1/30).



    You may use a pencil and paper. You may not use a calculator, computer, or logarithmic tables.
    Last edited by greg1313; Apr 24th 2019 at 01:21 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jun 2014
    From
    NJ
    Posts
    302
    Thanks
    116

    Re: Which radical expression is larger?

    Hint: try raising both expressions to the 30th power and see which is larger.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    Quote Originally Posted by ChipB View Post
    Hint: try raising both expressions to the 30th power and see which is larger.
    If you do that, then you will have 10^(15/11) versus 30. What would be your next
    step? I have it as a challenge problem, which means I'm challenging others.

    My strategy involves getting rid of fractional exponents for both sides at the outset.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Cervesa's Avatar
    Joined
    Dec 2014
    From
    USA
    Posts
    79
    Thanks
    63

    Re: Which radical expression is larger?

    $10^{1/22} < 30^{1/30}$

    Spoiler:
    $10^{15/11} \ne 3 \cdot 10$

    $10^{15} \ne 3^{11} \cdot 10^{11}$

    $10^4 \ne 3^3 \cdot 9^4$

    $\dfrac{10^4}{9^4} \ne 27$

    $\dfrac{10000}{(80+1)^2} \ne 27$

    $\dfrac{10000}{6400+160+1} < 2 < 27$
    Thanks from greg1313
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,990
    Thanks
    724

    Re: Which radical expression is larger?

    Quote Originally Posted by greg1313 View Post
    $\displaystyle \sqrt[22]{10} \ \ \ or \ \ \ \sqrt[30]{30}$

    The type is relatively small. That is the equivalent of 10^(1/22) versus 30^(1/30).



    You may use a pencil and paper. You may not use a calculator, computer, or logarithmic tables.
    \begin{align} \sqrt[22]{10} &\leftrightarrow \sqrt[30]{30} \\ \frac1{22}\log{10} &\leftrightarrow \frac1{30}\log{30} \\ 30\log{10} &\leftrightarrow 22(\log{10} + \log{3}) \\ 8\log{10} &\leftrightarrow 22\log{3} \\ \frac{\log{10}}{\log{3}} &\leftrightarrow \frac{22}{8} \\ 2 = \frac{2\log{3}}{\log{3}} = \frac{\log{9}}{\log{3}} \approx \frac{\log{10}}{\log{3}} &\lt \frac{22}8 \approx 3 \end{align}
    Last edited by Archie; Apr 25th 2019 at 07:18 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Nov 2018
    From
    USA
    Posts
    32
    Thanks
    4

    Re: Which radical expression is larger?

    $\displaystyle \frac{\sqrt[30]{30}}{\sqrt[22]{10}} = \frac{\sqrt[30]{10}\sqrt[30]{3}}{\sqrt[22]{10}} = \frac{\sqrt[150]{243}}{\sqrt[165]{100}} > 1 \longrightarrow \sqrt[30]{30} > \sqrt[22]{10}$
    Last edited by Zexuo; Apr 25th 2019 at 08:09 AM. Reason: Mistake
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    Quote Originally Posted by Zexuo View Post
    $\displaystyle \frac{\sqrt[30]{10}\sqrt[30]{3}}{\sqrt[22]{10}} = \frac{\sqrt[150]{243}}{\sqrt[165]{100}} $
    It looks like you're missing a step or two between these two.

    Quote Originally Posted by Zexuo View Post
    $\displaystyle \frac{\sqrt[150]{243}}{\sqrt[165]{100}} > 1 $
    No, I don't see how you have justified this part of it.
    Last edited by greg1313; Apr 25th 2019 at 10:30 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    Quote Originally Posted by Archie View Post
    \begin{align} \sqrt[22]{10} &\leftrightarrow \sqrt[30]{30} \\ \frac1{22}\log{10} &\leftrightarrow \frac1{30}\log{30} \\ 30\log{10} &\leftrightarrow 22(\log{10} + \log{3}) \\ 8\log{10} &\leftrightarrow 22\log{3} \\ \frac{\log{10}}{\log{3}} &\leftrightarrow \frac{22}{8} \\ 2 = \frac{2\log{3}}{\log{3}} = \frac{\log{9}}{\log{3}} \approx \frac{\log{10}}{\log{3}} &\lt \frac{22}8 \approx 3 \end{align}
    Archie, (in effect) you supported that 2 < $\displaystyle \ \dfrac{log(10)}{log(3)}, \ \ $ and we know that 2 = 16/8 < 22/8.

    But I do not see support as to where you show the relative size of $\displaystyle \ \dfrac{log(10)}{log(3)} \ $ versus 22/8.
    Last edited by greg1313; Apr 25th 2019 at 10:43 AM.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    *
    Last edited by greg1313; Apr 25th 2019 at 10:41 AM. Reason: Looking to delete a duplicate post.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    Nov 2018
    From
    USA
    Posts
    32
    Thanks
    4

    Re: Which radical expression is larger?

    $\displaystyle \frac{\sqrt[30]{10}}{\sqrt[22]{10}} = 10^{\frac{1}{30}-\frac{1}{22}} = 10^{-\frac{2}{165}} = 100^{-\frac{1}{165}}$

    $\displaystyle \sqrt[30]{3} = \left(243^{\frac{1}{5}}\right)^{\frac{1}{30}} = 243^{\frac{1}{150}}$

    The first inequality justified by the shallower root of a larger number in the numerator compared with the denominator (with both radicands greater than 1).
    Last edited by Zexuo; Apr 25th 2019 at 10:47 AM.
    Thanks from greg1313
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,990
    Thanks
    724

    Re: Which radical expression is larger?

    Quote Originally Posted by greg1313 View Post
    Archie, (in effect) you supported that 2 < $\displaystyle \ \dfrac{log(10)}{log(3)}, \ \ $ and we know that 2 = 16/8 < 22/8.

    But I do not see support as to where you show the relative size of $\displaystyle \ \dfrac{log(10)}{log(3)} \ $ versus 22/8.
    Well $\frac{\log{27}}{\log{3}}=3$ and $10$ is clearly much closer to $9$ than it is to $27$, so it is reasonable to assume that we have a number close to $2$. $\frac{22}{8}=2.75$ which is closer to $3$ than it is to $2$. With a bit of thought I could probably find something a little more rigorous, but it didn't seem necessary.

    E.g: The geometric mean of $9$ and $27$ is $\sqrt{243}$ which is a little larger than $15$. Since $10<15$ we know that $\frac{\log{10}}{\log{3}} < \frac{\log{15}}{\log{3}} < 2.5$
    Last edited by Archie; Apr 25th 2019 at 11:48 AM.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,990
    Thanks
    724

    Re: Which radical expression is larger?

    Or, perhaps better:
    $$\frac{\log{10}}{\log{3}} < \frac{\log{\sqrt{243}}}{\log{3}} = \frac{\log{243}}{2\log{3}} = \frac{\log{3^5}}{2\log{3}} = \frac{5\log{3}}{2\log{2}} = \frac52 < \frac{22}{8}$$
    Thanks from greg1313
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    Archie, your fifth denominator has a typo. It should be 2log3. You may have noticed
    that I gave that post a "Thanks."
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    275
    Thanks
    144

    Re: Which radical expression is larger?

    $\displaystyle 10^{1/22} \ \ vs. \ \ 30^{1/30}$

    $\displaystyle (10^{1/22})^{(2*11*15)} \ \ vs. \ \ (30^{1/30})^{(2*11*15)}$

    $\displaystyle 10^{15} \ \ vs. \ \ 30^{11}$

    $\displaystyle \dfrac{10^{15}}{10^{11}} \ \ vs. \ \ \dfrac{30^{11}}{10^{11}}$

    $\displaystyle 10^4 \ \ vs. \ \ \bigg(\dfrac{30}{10}\bigg)^{11}$

    $\displaystyle 10^4 \ \ vs. \ \ 3^{11}$

    $\displaystyle \dfrac{10^4}{3^{12}} \ \ vs. \ \ \dfrac{3^{11}}{3^{12}}$

    $\displaystyle \dfrac{10^4}{(3^3)^4} \ \ vs. \ \ \dfrac{1}{3}$

    $\displaystyle \bigg(\dfrac{10}{27}\bigg)^4 \ \ vs. \ \ \dfrac{1}{3}$

    $\displaystyle \bigg(\dfrac{10}{27}\bigg)^4 \ < \bigg(\dfrac{10}{20}\bigg)^4 \ = \ \bigg(\dfrac{1}{2}\bigg)^4 \ = \ \dfrac{1}{16} \ < \ \dfrac{1}{3}$


    Therefore, $\displaystyle \ \ 30^{1/30} \ > \ 10^{1/22} $
    Last edited by greg1313; Apr 27th 2019 at 09:33 AM.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Which log expression is larger?
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: Apr 7th 2019, 08:15 PM
  2. Replies: 1
    Last Post: Aug 25th 2017, 09:57 AM
  3. Radical Expression Help Please!
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: Feb 14th 2016, 07:31 PM
  4. Radical Expression
    Posted in the Algebra Forum
    Replies: 6
    Last Post: Apr 29th 2010, 06:21 PM
  5. Help with radical expression!!!
    Posted in the Algebra Forum
    Replies: 7
    Last Post: Sep 10th 2008, 06:31 AM

/mathhelpforum @mathhelpforum