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Simplify the square root

  1. Jan 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Simplify

    [tex]\sqrt {10 + \sqrt{24} + \sqrt{40}+\sqrt{60}}[/tex]


    2. Relevant equations
    [tex]\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}[/tex]


    3. The attempt at a solution
    [tex]\sqrt {10 + \sqrt{24} + \sqrt{40}+\sqrt{60}}[/tex]
    [tex]= \sqrt {10 + 2 \sqrt{6} + 2 \sqrt{10}+ 2 \sqrt{15}}[/tex]

    Stuck...
     
  2. jcsd
  3. Jan 19, 2012 #2

    SammyS

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    Why is it that [itex]\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}\,?[/itex]

    It's because

    [itex]a+b+2\sqrt{ab}=\sqrt{a}^2+2\sqrt{a}\sqrt{b}+\sqrt{b}^2[/itex]
    [itex]\displaystyle=\left(\sqrt{a}+\sqrt{b}\right)^2[/itex]​

    Now look at your final expression: [itex] \sqrt {10 + 2 \sqrt{6} + 2 \sqrt{10}+ 2 \sqrt{15}}\,.[/itex]

    What are the prime factors of 6? ... of 10? ... of 15 ?

    Expand [itex](x+y+z)^2\,.[/itex]
     
  4. Jan 19, 2012 #3

    ehild

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    SammyS, you are the greatest square-root simplifier!!!:biggrin:

    ehild
     
  5. Jan 19, 2012 #4
    Thanks :smile:

    I agree :biggrin::approve:
     
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