Q&A for How to Rationalize the Denominator

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  • Question
    How do I rationalize with three terms?
    Community Answer
    Something like 1/(1+root2 + root3)? If so, group as 1+(root2 + root3) and multiply through by the "difference of squares conjugate" 1-(root2 + root3). That makes the denominator -4 - root6, which is still irrational, but did improve from two irrational terms to only one. So repeat the same trick by multiplying through by -4+root6 and the denominator is rationalized.
  • Question
    In your pictures, what does the point mean?
    Donagan
    Top Answerer
    If you're asking about the dots that are placed between various fractions, those are multiplication signs. For example, in the article's second image we see (7√3) / (2√7), then a dot, then (√7 / √7). That means we multiply the first fraction by the second fraction (numerator times numerator, and denominator times denominator), giving us (7√21) / 14, which simplifies to √21 / 2. (Incidentally, the article shows some other dots that are not between fractions. Those are merely "bullet points.")
  • Question
    How can I rationalize the denominator with a cube root that has a variable?
    Community Answer
    If it is a binomial expression, follow the steps outlined in method 2.
  • Question
    How do you rationalize a cube root in the denominator for a question like 1/(cube root 5- cube root 3)?
    Community Answer
    This is a little trickier, but can be done. Multiply top and bottom by (cuberoot 25 + cuberoot 15 + cuberoot 9) and the denominator simplifies to 2. This trick is analogous to the quadratic case since it uses the difference of cubes factorization of 5-3, whereas the quadratics use the difference of squares factorization.
  • Question
    How do you rationalize the denominator of 1 / √11 - 3?
    Community Answer
    Multiply both the numerator and denominator by the conjugate of the denominator, which is √11 + 3. The expression becomes (√11 + 3) / 11 - 9, or (3 + √11) / 2.
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