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Math Help - Help with Surds

  1. #1
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    Help with Surds

    Hi,

    I need a bit of help solving a question on surds if anyone can push me in the right direction with it.

    The question is:


    I've started multiplying out the bottom row, and have got as far as this:

    \frac{4\sqrt{3}+3\sqrt{7}}{3\sqrt{3}+\sqrt{7}} * \frac{3\sqrt{3}-\sqrt{7}}{3\sqrt{3}-\sqrt{7}}

    I then proceed to multiplying out the bottom row;

    3*3+3\sqrt{3}-3\sqrt{7}+3\sqrt{3}+\sqrt{3}*\sqrt{3}-\sqrt{21}+3\sqrt{7}+\sqrt{21}-7

    I then proceed to collecting the like terms;

    (9+3-7)+(-3\sqrt{7}+3\sqrt{7})+(-\sqrt{21}+\sqrt{21})+(3\sqrt{3}+3\sqrt{3})

    = 5 + 6\sqrt{3}

    From here, I really can't see how to 'rationalise the denominator'.

    I'm only working on the bottom row right now, but if anyone can tell me where I'm going wrong with this that would be great. I'm sure it's something simple, I'm just not seeing it.

    Thanks in advance.
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  2. #2
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    Re: Help with Surds

    Quote Originally Posted by ashleysmithd View Post
    Hi,

    I need a bit of help solving a question on surds if anyone can push me in the right direction with it.

    The question is:


    I've started multiplying out the bottom row, and have got as far as this:

    \frac{4\sqrt{3}+3\sqrt{7}}{3\sqrt{3}+\sqrt{7}} * \frac{3\sqrt{3}-\sqrt{7}}{3\sqrt{3}-\sqrt{7}}

    I then proceed to multiplying out the bottom row;

    ...
    Not sure what you've done in the next steps.

    Use (a + b)(a - b) = aČ - bČ

    That means the denominator becomes:

    ({3\sqrt{3}+\sqrt{7}}) \cdot ({3\sqrt{3}-\sqrt{7}})=9\cdot 3 - 7 =20

    then:
    - expand the numerator
    - collect those terms with \sqrt{21} and those with simple integers
    - factor out 5 at numerator and denominator
    - cancel the common factor

    You should come out with \frac{3+\sqrt{21}}4
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  3. #3
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    Re: Help with Surds

    Thanks for the reply, this is starting to make sense.

    The only confusion I still have is how multiplying out the numerator becomes 15+\sqrt{21}

    (4*3)*3 = 36
    (3\sqrt{7}*-\sqrt{7}) = -21
    \Rightarrow 36 - 21 to me is making 15, only.

    In what order is it multiplied out in to make 15+\sqrt{21}?

    Thanks again.
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  4. #4
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    Re: Help with Surds

    So you need to expand this guy?

    (4\sqrt{3}+3\sqrt{7})(3\sqrt{3}-\sqrt{7})

    Follow this method (a+b)(c+d) = ac+ad+bc+bd

    Does this make sense?
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  5. #5
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    Re: Help with Surds

    Ah, yes. I was trying to use the (a+b)(c+d) = ac + ad + bc + bd before except I was separating the coefficients from the roots, completely, hence why my multiplying out was so long because I was multiplying through several times too many. Basically, I didn't understand that 3\sqrt{7} was a single term, I was treating the 3 and \sqrt{7} as if they were separate.

    Many thanks.

    There's just one last thing I don't get... For the numerator, I got 15+5\sqrt{21}. Where does the 5 go? I know earboth said about factoring it out at the numerator and denominator, but I'm not sure I understand.
    Last edited by ashleysmithd; January 5th 2012 at 02:34 PM.
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  6. #6
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    Re: Help with Surds

    15+5\sqrt{21} = 5\times 3 +5\times \sqrt{21} =  5(3 + \sqrt{21}) now what did you get for the denominator?
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  7. #7
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    Re: Help with Surds

    I got 20 for the denominator, I worked this out through;

    (3\sqrt{3}+\sqrt{7})(3\sqrt{3}-\sqrt{7})

    = 9\sqrt{9} -3\sqrt{21}+3\sqrt{21}-7
    = 9(3)-7
    = 27 - 7 = 20

    I'm still a bit unsure about the last bit, where the 5 goes. I can understand that 15+5\sqrt{21} is equivalent to 5(3+\sqrt{21}), but multiplied out that still gives 15+5\sqrt{21}.

    Apologies, I really don't get this last bit.
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  8. #8
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    Re: Help with Surds

    Ok, you have done all the hard work here, just need to cancel some terms,

    \frac{15+5\sqrt{21}}{20} = \frac{5\times 3 +5\times \sqrt{21}}{5\times 4} =  \frac{5(3 + \sqrt{21})}{5\times 4} =  \frac{3 + \sqrt{21}}{4}


    as given in post #2.
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  9. #9
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    Re: Help with Surds

    Ahh.. I understand now. It's just simplifying the fraction ultimately.

    Many thanks pickslides and earboth.
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