Q&A for How to Rationalize the Denominator

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How do I rationalize with three terms?

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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.

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In your pictures, what does the point mean?

Donagan

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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.")

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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.

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How do you rationalize a cube root in the denominator for a question like 1/(cube root 5- cube root 3)?

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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.

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How do you rationalize the denominator of 1 / √11 - 3?

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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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