Rationalizing the Denominator
Two meanings of 'rationalizing the denominator': the algebra move, and making excuses for skipping it.
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Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2021-01-01-rationalizing-the-denominator/ (CC BY-NC 4.0)Transcript
[The Denominator] 1/√2 (denominator circled)
[Rationalizing the Denominator (version 1)] 1/√2 × √2/√2 = √2/2 ("rational!")
[Rationalizing the Denominator (version 2)] A student: "It's fine like it is, right? Does anyone REALLY care where the radical is?"