Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Simplify and classify a concrete product of irrational expressions
Problem
Evaluate \(\sqrt{2}\cdot~\sqrt{2}\) using the square-root product identity, then classify the exact product.
Big Picture
What this problem is really about
Multiplying a principal square root by itself returns its radicand, so irrational factors need not produce an irrational product. We’ll apply that identity, verify it through the radical product property, classify the exact integer result, and state what the example shows about closure.
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