Rationalize a fraction.Rationalize the denominator 1 2 1 3 + 3 1 3 + 4 1 3 . Is there a short solution for this task ? Thanks in advance.

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Rationalize a fraction.Rationalize the denominator       1                  2                              1            3                              +              3                              1            3                              +              4                              1            3                              . Is there a short solution for this task ? Thanks in advance.

Blaine Day · Accepted Answer

First, use the identity  (  a  +  b  +  c  )  (      a    2    +      b    2    +      c    2    −  a  b  −  a  c  −  b  c  )      =          a    3    +      b    3    +      c    3    −  3  a  b  cwith   a  =      2    3   and   b  =      3    3   and   c  =      4    3    .Multiplying both the numerator and denominator of your fraction by the numerical form of       a    2    +      b    2    +      c    2    −  a  b  −  a  c  −  b  c will give you      stuff          2      +      3      +      4      −      3              24        3                  =          stuff          9      −      6              3        3            Now use the identity  (  a  −  b  )  (      a    2    +  a  b  +      b    2    )  =      a    3      –        b    3  with   a  =  9 and   b  =  6      3    3    .Multiplying both the numerator and denominator of the displayed fraction above by the numerical form of       a    2    +  a  b  +      b    2   will give you            (      stuff      )      (      other stuff      )                      9        3            −      (      6              3        3                    )        3                  =                (      stuff      )      (      other stuff      )        81  At this point you simply need to multiply out (stuff)(other stuff). This is a bit tedious, but not really excessively so --- it&#039;ll be polynomial with 6 terms multiplied by a polynomial with 3 terms, for a total of 18 &quot;FOIL multiplications&quot; that need to be carried out.

Rationalize a fraction. Rationalize the denominator (1)/(2^((1)/(3)) + 3^((1)/(3)) + 4^((1)/(3)). Is there a short solution for this task ? Thanks in advance.

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