What is the math question that can be stated in two seconds but is most often answered incorrectly by math majors? In my opinion, here it is:

Simplify .

In my normal conversational voice, I can say “Simplify the square root of squared” in a shade less than two seconds.

Here’s a thought bubble if you’d like to think about this before I give the answer.

A common mistake made by algebra students (and also math majors in college who haven’t thought about this nuance for a while) is thinking that . This is clearly incorrect if is negative:

The second follow-up mistake is then often mistake made by attempting to rectify the first mistake by writing . The student usually intends the symbol to mean “plus or minus, depending on the value of ,” whereas the true meaning is “plus or minus” without any caveats. I usually correct this second mistake by pointing out that when a student finds with a calculator, the calculator doesn’t flash between and ; it returns only one answer.

After clearing that conceptual hurdle, students can usually guess the correct simplification:

In this series of posts, I’d like to expand on the thoughts above to consider some of the inverse functions that commonly appear in secondary mathematics: the square-root function and the inverse trigonometric functions.

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