To determine which product yields an irrational number, let's evaluate each option step by step:
A. \( \sqrt{9} \times 3 \) \[ \sqrt{9} = 3 \quad \Rightarrow \quad 3 \times 3 = 9 \] (9 is a rational number)
B. \( \frac{7}{8} \times \sqrt{25} \) \[ \sqrt{25} = 5 \quad \Rightarrow \quad \frac{7}{8} \times 5 = \frac{35}{8} \] (\(\frac{35}{8}\) is a rational number)
C. \( 2\sqrt{64} \times 7\sqrt{81} \) \[ \sqrt{64} = 8 \quad \text{and} \quad \sqrt{81} = 9 \quad \Rightarrow \quad 2 \times 8 \times 7 \times 9 = 2 \times 8 \times 63 = 1008 \] (1008 is a rational number)
D. \( 4\sqrt{5} \times 2\sqrt{49} \) \[ \sqrt{49} = 7 \quad \Rightarrow \quad 4\sqrt{5} \times 2 \times 7 = 8\sqrt{5} \] (8\(\sqrt{5}\) is an irrational number, since \(\sqrt{5}\) is not a perfect square)
Based on this evaluation:
- A yields a rational number.
- B yields a rational number.
- C yields a rational number.
- D yields an irrational number.
Thus, the answer is: D. \( 4\sqrt{5} \times 2\sqrt{49} \) yields an irrational number.
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