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ParthKohli Group Title

Denesting radicals. How to denest \(\sqrt{2 + \sqrt{2}}\)

  • one year ago
  • one year ago

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  1. ParthKohli Group Title
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    \[\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}\]I'm trying to do this problem.

    • one year ago
  2. asnaseer Group Title
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    one way is to write it as:\[x=\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+...}}}}}\]and then square both sides to get:\[x^2=2+\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+...}}}}}=2+x\]then just solve the quadratic

    • one year ago
  3. asnaseer Group Title
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    unless I have mis understood your question?

    • one year ago
  4. ParthKohli Group Title
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    Actually, I'd just want to know how to denest \(\sqrt{2 + \sqrt2}\)

    • one year ago
  5. asnaseer Group Title
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    I guess you could use the same technique, i.e:\[x=\sqrt{2+\sqrt{2}}\]\[x^2=2+\sqrt{2}\]\[x^2-2=\sqrt{2}\]then square both sides and solve the resulting quartic equation

    • one year ago
  6. asnaseer Group Title
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    which should end up being just a quadratic in \(x^2\)

    • one year ago
  7. ParthKohli Group Title
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    Ah,\[(x^2 - 2)^2 = 2\]\[x^4 -4x^2 +4 = 2 \]\[x^4 - 4x^2 - 2 = 0\]Assume \(y = x^2\).\[y^2 - 4y - 2 = 0\]\[y = 2\pm \sqrt{6}\]\[x = \sqrt{2 \pm \sqrt6}\]

    • one year ago
  8. ParthKohli Group Title
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    @asnaseer But that again brings up a nested radical.

    • one year ago
  9. asnaseer Group Title
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    I'm not sure what exactly you mean by "denest" here?

    • one year ago
  10. ParthKohli Group Title
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    Well, a radical expression containing another radical expression.

    • one year ago
  11. asnaseer Group Title
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    btw: you have a small sign error in your calculation above

    • one year ago
  12. ParthKohli Group Title
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    Oh, that was supposed to be a +... not -.

    • one year ago
  13. asnaseer Group Title
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    are you trying to simply \(\sqrt{2+\sqrt{2}}\) further?

    • one year ago
  14. ParthKohli Group Title
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    A second please.

    • one year ago
  15. ParthKohli Group Title
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    Yeah, I want to remove the inner radical.

    • one year ago
  16. asnaseer Group Title
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    why?

    • one year ago
  17. ParthKohli Group Title
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    http://math.stackexchange.com/questions/196155/strategies-to-denest-nested-radicals I can't get Bill Dubuque's technique to work here. The motivation behind this question is... I want to learn something new. :)

    • one year ago
  18. asnaseer Group Title
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    so you are trying to express this as:\[\sqrt{2+\sqrt{2}}=a\sqrt{b}\]and what to find a and b?

    • one year ago
  19. ParthKohli Group Title
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    Yeah, I want this in the form \(a + b\sqrt n \), not \(\sqrt{a + b\sqrt n }\)

    • one year ago
  20. asnaseer Group Title
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    ok, I see now - let me think over this. I am at work now so will probably get back to this after work - looks like an interesting problem. :)

    • one year ago
  21. ParthKohli Group Title
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    Thank you!\[2 + \sqrt 2\]Here a = 2, b = 1, n = 2. So,\[\sqrt{\text{parth}} = \sqrt{a^2 - nb^2 } = \sqrt{4 - 2} = \sqrt{2}\]Subtracting the above,\[2 + \sqrt{2} - \sqrt{2} = 2\]Now, find \(\sqrt{\text{kohli} } = \sqrt{2a} = \sqrt{4} = 2\). But that yields \(1\).

    • one year ago
  22. mathslover Group Title
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    • one year ago
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  23. mathslover Group Title
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    answer = 1

    • one year ago
  24. mathslover Group Title
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    sorry ..

    • one year ago
  25. ParthKohli Group Title
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    Is 1 correct?

    • one year ago
  26. Ishaan94 Group Title
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    Of course, NOT!

    • one year ago
  27. asnaseer Group Title
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    ok, after some research, I found out that this is actually quite a difficult problem. however, there are some special cases that can be solved for quite easily. see this wiki article for details: http://en.wikipedia.org/wiki/Nested_radical the article mentions: "In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested. Earlier algorithms worked in some cases but not others."

    • one year ago
  28. sauravshakya Group Title
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    WELL:\[\sqrt{2+\sqrt{2}} \] cannot be expressed in the forms |dw:1348223973261:dw|

    • one year ago
  29. ParthKohli Group Title
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    Thank God! I was so confused.

    • one year ago
  30. sauravshakya Group Title
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    But I think it can be expressed in some form like|dw:1348224135337:dw|

    • one year ago
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