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Math Help - Rationalise the denominator and simplify

  1. #1
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    Smile Rationalise the denominator and simplify

    1/2√7 + 2√3 - 1/3√7 + √3
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  2. #2
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    Is that \frac{1}{2\sqrt{7}} + 2\sqrt{3} - \frac{1}{3\sqrt{7}} + \sqrt{3}

    or  \frac{1}{2} \sqrt{7} + 2\sqrt{3} - \frac{1}{3} \sqrt{7} + \sqrt{3} ?

    I'll assume its the first one since we are supposed to rationalise a denominator.

    \frac{1}{2\sqrt{7}} + 2\sqrt{3} - \frac{1}{3\sqrt{7}} + \sqrt{3}

    =\frac{3 - 2 +3\sqrt{3} \times 6\sqrt{7}}{6\sqrt{7}}

    =\frac{1 +18\sqrt{21} }{6\sqrt{7}}

    Now times top and bottome by \sqrt{7} to rationalise the denominator.
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  3. #3
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    easy.....

    I think it's :
    \frac {1} {2\sqrt{7} + 2\sqrt {3}} - \frac {1} {3\sqrt {7} + \sqrt {3}}

    right??
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  4. #4
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    Sorry, I meant:

    1/(2√7 + 2√3) - 1/(3√7 + √3)

    LOL, I was thinking of putting brackets but hesitated.
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  5. #5
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    Quote Originally Posted by sudha_14 View Post
    I think it's :
    \frac {1} {2\sqrt{7} + 2\sqrt {3}} - \frac {1} {3\sqrt {7} + \sqrt {3}}

    right??
    Yep!
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  6. #6
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    \frac {1} {2\sqrt{7} + 2\sqrt {3}} - \frac {1} {3\sqrt {7} + \sqrt {3}}

    = \frac{3\sqrt {7} + \sqrt {3} - 2\sqrt{7} - 2\sqrt{3}}{(2\sqrt{7} + 2\sqrt {3})(3\sqrt {7} + \sqrt {3})}

    = \frac{\sqrt {7} - \sqrt {3}}{(2\sqrt{7} + 2\sqrt {3})(3\sqrt {7} + \sqrt {3})}

    Now to rationalise the denominator times the fraction (bottom and top) by the conjugates of the two surds to make the two factors rational.

    = \frac{(\sqrt {7} - \sqrt {3})(2\sqrt{7} - 2\sqrt {3})(3\sqrt {7} - \sqrt {3})}{(2\sqrt{7} + 2\sqrt {3})(2\sqrt {7} - 2\sqrt {3})(3\sqrt{7} + \sqrt {3})(3\sqrt {7} - \sqrt {3})}

    = \frac{(\sqrt {7} - \sqrt {3})(2\sqrt{7} - 2\sqrt {3})(3\sqrt {7} - \sqrt {3})}{(28 - 12)(63 - 3)}

    = \frac{(\sqrt {7} - \sqrt {3})(2\sqrt{7} - 2\sqrt {3})(3\sqrt {7} - \sqrt {3})}{16 \times 60}

    = \frac{(\sqrt {7} - \sqrt {3})(2\sqrt{7} - 2\sqrt {3})(3\sqrt {7} - \sqrt {3})}{960}

    Now, I leave it up to you to expand the top...
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  7. #7
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    pretty fast....ur reply i mean....

    i'm so slow at typing the math tag....
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