Why do we rationalize the denominator?


Our math teachers always tell us to "rationalize the denominator", but most of the time they don't tell us why. I'll leave it up to you to decide whether or not you think the reasons for rationalizing are good ones, but here are some of the reasons why we do it.

0:13 What we mean when we say "rationalize the denominator" // We're basically just saying "get the root out of the denominator". To do this, we have to multiply both the numerator and denominator by the root that's in the denominator. That way, the roots will cancel. Pro tip: If you have more than just a single root in your denominator, try conjugate method to eliminate the root from the denominator.

1:28 Why should be bother rationalizing the denominator at all? // We'll go over a few reasons why it might be a good idea:

1:34 It's easier for teachers to grade work when everyone's giving their answers in the same format.

1:51 Historically, we think about rationalized fractions as being reduced to lower terms, compared with non-rationalized fractions. And we always want to have our fractions reduced to the lowest possible terms. The reason rationalized fractions are in lower terms is because, before we had calculators, it was easier to do the long division for a rationalized fraction, than it was to do the long division for a non-rationalized fraction.

2:35 Agreeing to rationalize our fractions means we can always recognize like-terms when we have them, which might help us further simplify our answers.

3:00 Summary

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