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Math Help - Simplifying this fraction

  1. #1
    Newbie
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    Simplifying this fraction

    My apologies.

    64a^3 + b^3
    ___________
    16 a^2 - b^2

    Divided by

    16a^2b^2 - 4ab^3 + b^4
    _______________________
    4a^2 - ab + 12a - 3b

    Sorry I don't know how to use the math function yet.
    Last edited by ThePerfectHacker; June 9th 2006 at 11:11 AM.
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  2. #2
    Super Member

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    Hello, sstr!

    A fascinating problem: four types of factoring are required.


    \frac{64a^3 + b^3}{16a^2 - b^2} \div \frac{16a^2b^2 - 4ab^3 + b^4}{4a^2 - ab + 12a - 3b}

    Sum of Cubes: . 64a^3 + b^3 \;= \;(4a + b)(16a^2 - 4ab + b^2)

    Difference of Squares: . 16a^2 - b^2 \;= \;(4a - b)(4a + b)

    Common factors: . 16a^2b^2 - 4ab^3 + b^4 \;= \;b^2(16a^2 - 4ab + b^2)

    Grouping: . 4a^2 - ab + 12a - 3b \;= \;a(4a - b) + 3(4a - b)\;= \;(4a-b)(a+3)


    The problem becomes: . \frac{(4a + b)(16a^2 - 4ab + b^2)}{(4a - b)(4a + b)}\; \times  \frac{(4a - b)(a + 3)}{b^2(16a^2 - 4ab + b^2)}

    . . . which reduces to: . \frac{a + 3}{b^2}

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