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Math Help - Simplifying an Expression with Radicals

  1. #1
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    Simplifying an Expression with Radicals

    Can any of you simplify this expression?

    Simplifying an Expression with Radicals-img_20121012_101703-1.jpg

    Thank you!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Simplifying an Expression with Radicals

    Yes, by inspection it's \sqrt{2}+1
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  3. #3
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Simplifying an Expression with Radicals

    Quote Originally Posted by jigogwapo16 View Post
    Can any of you simplify this expression?
    Click image for larger version. 

Name:	IMG_20121012_101703-1.jpg 
Views:	17 
Size:	131.2 KB 
ID:	25179Thank you!

    Real & imaginary parts of the ratio for
    n=1,2,3,....,100 :: Real
    n=101,102,...,300 :: Imaginary

    Attached Thumbnails Attached Thumbnails Simplifying an Expression with Radicals-simplify.png  
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    Re: Simplifying an Expression with Radicals

    How do I solve this analytically using algebra? Or would it be impossible?

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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: Simplifying an Expression with Radicals

    Sorry for the delay, I had to ask my girlfriend how to do this...

    We are given:

    \frac{\sum_{k=1}^{99}(\sqrt{10+\sqrt{k}})}{\sum_{k  =1}^{99}(\sqrt{10-\sqrt{k}})}

    Let:

    r=\sqrt{10+\sqrt{k}}-\sqrt{10-\sqrt{k}}

    r^2=10+\sqrt{k}-2\sqrt{100-k}+10-\sqrt{k}

    r^2=2(10-\sqrt{100-k})

    Since 0<r we may write:

    r=\sqrt{2}\sqrt{10-\sqrt{100-k}}

    Hence, we may write:

    \frac{\sum_{k=1}^{99}(\sqrt{10+\sqrt{k}})}{\sum_{k  =1}^{99}(\sqrt{10-\sqrt{k}})}=

    \frac{\sum_{k=1}^{99}(r+\sqrt{10-\sqrt{k}})}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}=

    \frac{\sum_{k=1}^{99}(r)+\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}=

    \frac{\sum_{k=1}^{99}(r)}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}+\frac{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}=

    \frac{\sum_{k=1}^{99}(\sqrt{2}\sqrt{10-\sqrt{100-k}})}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}+1=

    \sqrt{2}\frac{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}{\sum_{k=1}^{99}(\sqrt{10-\sqrt{k}})}+1=

    \sqrt{2}+1
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  6. #6
    Junior Member alane1994's Avatar
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    Re: Simplifying an Expression with Radicals

    ^^^^hhhooolllyyycccrrraaappp.... mind... blown...
    Also, Do you think they know summation notation?
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: Simplifying an Expression with Radicals

    Given the expression the OP was asked to simplify, I would assume knowledge of summation or sigma notation.
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  8. #8
    Junior Member alane1994's Avatar
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    Re: Simplifying an Expression with Radicals

    Very true.
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  9. #9
    Junior Member alane1994's Avatar
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    Re: Simplifying an Expression with Radicals

    Confused me...
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