Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Classify a rational-irrational product with an explicit zero check
Problem
Simplify and classify \(3\sqrt{2}\) as rational or irrational. Verify that the rational factor is nonzero and use that fact in the justification.
Big Picture
What this problem is really about
The rational-times-irrational rule needs an explicit nonzero condition. We’ll classify both factors, check that the rational factor is not zero, apply the qualified rule, and verify it by showing that a rational product would force the known irrational factor to become rational after division.
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