Results 1 to 5 of 5

Math Help - [SOLVED] Simplifying an expression with radicals

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    2

    [SOLVED] Simplifying an expression with radicals

    The problem is the following:

    \sqrt{8+2\sqrt{10+2\sqrt{5}}}+\sqrt{8-2\sqrt{10+2\sqrt{5}}}

    I have to simplify it, but I have no idea how to start. Any help is appreciated. Thanks!

    Pharod
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Hello Pharod

    Welcome to Math Help Forum!
    Quote Originally Posted by Pharod View Post
    The problem is the following:

    \sqrt{8+2\sqrt{10+2\sqrt{5}}}+\sqrt{8-2\sqrt{10+2\sqrt{5}}}

    I have to simplify it, but I have no idea how to start. Any help is appreciated. Thanks!

    Pharod
    Let x = \sqrt{a+b}+\sqrt{a-b}, where a = 8 and b = 2\sqrt{10+2\sqrt5}

    Then x^2 = a+b+2\sqrt{a^2-b^2}+a-b
    =2a+2\sqrt{a^2-b^2}

    =16+2\sqrt{24-8\sqrt5}

    =4(4+\sqrt{6-2\sqrt5})
    Also 6-2\sqrt5 = (\sqrt5 - 1)^2

    \Rightarrow \sqrt{6-2\sqrt5} = \sqrt5 - 1

    \Rightarrow x^2 = 4(4+\sqrt5-1)

    \Rightarrow x = 2\sqrt{3+\sqrt5}

    I don't think it will simplify any more.

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Grandad View Post

    \vdots
    \Rightarrow x = 2\sqrt{3+\sqrt5}

    I don't think it will simplify any more.
    It will go a bit further: (1+\sqrt5)^2 = 6+2\sqrt5, and so 2\sqrt{3+\sqrt5} = \sqrt2\sqrt{6+2\sqrt5} = \sqrt2(1+\sqrt5) = \sqrt2+\sqrt{10}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2010
    Posts
    2
    Hey, thank you both for leading me. I knew it had to be something with squaring, I just couldn't get to it for some reason.

    Pharod
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,553
    Thanks
    542
    Hello, Grandad!

    x \:=\: 2\sqrt{3+\sqrt5}

    I don't think it will simplify any more.

    3 + \sqrt{5} \:=\:\frac{(1 + \sqrt{5})^2}{2} \;=\;2\cdot\frac{(1+\sqrt{5})^2}{4}

    . . . . . .  \;=\;2\left(\underbrace{\frac{1+\sqrt{5}}{2}}_{\te  xt{Golden Mean}}\right)^2 \;=\;2\,\phi^2


    Therefore: . x \;=\;2\sqrt{2}\,\phi

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simplifying Radicals
    Posted in the Algebra Forum
    Replies: 5
    Last Post: August 29th 2011, 07:08 AM
  2. [SOLVED] Simplifying Radicals
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 5th 2010, 06:34 AM
  3. [SOLVED] simplifying an expression
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 31st 2010, 07:03 PM
  4. [SOLVED] Simplifying the boolean expression
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 3rd 2009, 10:29 AM
  5. [SOLVED] Simplifying Radicals
    Posted in the Algebra Forum
    Replies: 2
    Last Post: December 13th 2006, 05:52 AM

Search Tags


/mathhelpforum @mathhelpforum