Results 1 to 3 of 3

Math Help - rational number

  1. #1
    Newbie
    Joined
    Jul 2009
    Posts
    9

    rational number

    prove \sqrt{7}/^{3}\sqrt{15} is rational


    thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by elmo View Post
    prove \sqrt{7}/^{3}\sqrt{15} is rational


    thanks in advance.
    I'm assuming you mean irrational?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by elmo View Post
    prove \sqrt{7}/^{3}\sqrt{15} is \color{red}\text{ir} \text{rational}.


    thanks in advance.
     a = \frac{\sqrt{7}}{\sqrt[3]{15}} = \sqrt[6]{\frac{7^3}{15^2}}

    Assume  a=\frac{p}{q} , where  (p,q)=1 .

    Then  a^6 = \frac{p^6}{q^6} .
    So  7^3\cdot q^6 = 15^2\cdot p^6 .

    This means  7^3\mid p^6 \implies 7\mid p since  7 is prime.
    So  15^2\cdot p^6 =15^2\cdot (7k)^6 = 7^6\cdot 15^2\cdot k^6 = 7^3\cdot q^6\implies 7^3\cdot 15^2\cdot k^6 = q^6 .

    Therefore  7 \mid q for the same reasons as above.
    But this means  7\mid (p,q) which is a contradiction since we were under the assumption that  (p,q)=1 .

    Therefore  \frac{\sqrt{7}}{\sqrt[3]{15}} is irrational.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Is this a rational number?
    Posted in the Algebra Forum
    Replies: 7
    Last Post: October 6th 2011, 09:40 AM
  2. Replies: 4
    Last Post: April 28th 2011, 07:20 AM
  3. [SOLVED] Rational Number
    Posted in the Algebra Forum
    Replies: 8
    Last Post: August 28th 2010, 05:33 AM
  4. Rational number
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: October 24th 2009, 12:12 AM
  5. Replies: 5
    Last Post: October 7th 2008, 01:55 PM

Search Tags


/mathhelpforum @mathhelpforum