What Makes a Surd — Rational, Irrational, and the Rules

Distinguish rational from irrational numbers, test whether a root is a surd by checking the radicand for a perfect power, apply the product rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and quotient rule $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$, and explain with a counter-example why $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$.

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