What is the "AHA!" moment?

Difficulty: Medium.

And now for yet another absolute value question. Seems like I had one just the other day.

Difficulty: Medium - Hard.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 1990

answer 37, answer 38, Published 5/24.

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