Generated by claude-sonnet-4-6 · 2026-06-15 10:29 · grounding rag
Model Answer
Answer: C — $(\sqrt{3},\ \sqrt{27})$
$\sqrt{3} \times \sqrt{27} = \sqrt{81} = 9$, which is rational. Both $\sqrt{3}$ and $\sqrt{27}$ are irrational numbers whose product is rational.
Explanation
- Options A and D: $\sqrt{16}=4$ and $\sqrt{36}=6$ are rational, so those pairs don't qualify.
- Option B: $\sqrt{5} \times \sqrt{2} = \sqrt{10}$, which is irrational.
- Option C: $\sqrt{3} \times \sqrt{27} = \sqrt{3 \times 27} = \sqrt{81} = 9$ ✓ — both factors are irrational, product is rational. This is the key check examiners expect you to perform quickly.