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Realizing both 15-4√14 and 15+4√14 are perfect squares was the unlock for me. Rewrite each radicand as (a-b)^2 and (a+b)^2 where ab=14 and a^2+b^2=15. Took me a minute to spot it.
Exactly, once you see 14 = 7·2 and 15 = 7+2 it just falls apart. Classic hidden-perfect-square setup.
Middle difficulty for me. I tried squaring the whole thing first and got buried in cross terms. Remembering (x+y)^2 = x^2 + y^2 + 2xy cleans it up fast.
Same, the cross term is what scared me until I noticed the product under the root simplifies nicely.
I almost picked 30 because I dropped the cross term. Good reminder to keep all three parts.
Can someone explain why the inner radicals don't need absolute values when we take the square root? I keep second-guessing that step.
Both radicands are positive and so are the simplified bases, so no sign issue here. But check each factor is positive before you drop the absolute value.
A bit of a trap if you jump to the answer choices. The perfect-square recognition is the whole game. Solid 600-level question.
What it tests
Your handling of square and higher roots, including simplifying radicals.
Common trap
Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.