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Math Help - Algebraic addition/simplification prob.

  1. #1
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    Algebraic addition/simplification prob.

    Simplify,

    \frac{a}{ab+{b}^{2 }}+\frac{b}{{a}^{2}+ab}

    I get the principle i.e. multiply denominators to find common denominator, add, factor, simplify. I've done loads of examples but for some reason I just can't get this one. I know it's relatively straight forward, I think my problems somewhere in the simplification/indices area.

    I get as far as:-

    =\frac{(a)({a}^{2}+ab) + (b)(ab+{b}^{2})}{(ab+{b}^{2})({a}^{2}+ab)}

    =\frac{({a}^{3}+{a}^{2}b)+(a{b}^{2}+{b}^{3})}{(ab+  {b}^{2})({a}^{2}+ab)}

    At this point it all seems to get overly complicated.
    Anybody break it down a bit for me?
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  2. #2
    A Plied Mathematician
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    Always look to see if you can simplify before plowing ahead with fractions. You didn't do anything wrong, but you're making life a bit harder for yourself than you need to. Here's what I would do:

    \frac{a}{ab+b^{2}}+\frac{b}{a^{2}+ab}=\frac{a}{b(a  +b)}+\frac{b}{a(a+b)}.

    Does that make things a bit easier?
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  3. #3
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    Thanks, this was one of the many roads I started down, but never quite got to the end of. I can at least, get to the answer now,

    \frac{{a}^{2}+{b}^{2}}{ab(a+b)}

    but remain a bit unconvinced by my methodology.

    Am I right in thinking

    b(a+b) + a(a+b) = A common denominator ab(a+b)

    In which case I only need to muliply each numerator by the 'missing' bit of the denominator, i.e.

    \frac{a}{b(a+b)} x \frac{a}{a} = \frac{{a}^{2} }{ab(a+b)}

    and the same again for the other half?
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  4. #4
    A Plied Mathematician
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    Exactly right. Be convinced!
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