Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Simplify and classify a concrete sum of irrational expressions
Problem
Combine the additive inverses in \(\sqrt{2}+(-\sqrt{2})\), then classify the simplified value as rational or irrational.
Big Picture
What this problem is really about
Two irrational terms can have a rational sum when their irrational parts cancel. We’ll recognize the additive-inverse structure, combine the radical coefficients, simplify the result exactly, classify that value as an integer ratio, and use it to reject an invalid blanket closure claim.
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