Let's analyze each option to determine if it yields an irrational number:
A. \( \sqrt{9} \cdot 3 \)
Calculating \( \sqrt{9} \): \[ \sqrt{9} = 3 \] Thus, \[ \sqrt{9} \cdot 3 = 3 \cdot 3 = 9 \] This is a rational number.
B. \( \frac{7}{8} \cdot \sqrt{25} \)
Calculating \( \sqrt{25} \): \[ \sqrt{25} = 5 \] Thus, \[ \frac{7}{8} \cdot 5 = \frac{35}{8} \] This is also a rational number.
C. \( 2\sqrt{64} \cdot 7\sqrt{81} \)
Calculating \( \sqrt{64} \) and \( \sqrt{81} \): \[ \sqrt{64} = 8 \quad \text{and} \quad \sqrt{81} = 9 \] Thus, \[ 2\sqrt{64} = 2 \cdot 8 = 16 \] \[ 7\sqrt{81} = 7 \cdot 9 = 63 \] Now calculating \( 16 \cdot 63 \): \[ 16 \cdot 63 = 1008 \] This is a rational number.
D. \( 4\sqrt{5} \cdot 2\sqrt{49} \)
Calculating \( \sqrt{49} \): \[ \sqrt{49} = 7 \] Thus, \[ 2\sqrt{49} = 2 \cdot 7 = 14 \] Now calculating \( 4\sqrt{5} \cdot 14 \): \[ 4\sqrt{5} \cdot 14 = 56\sqrt{5} \] Since \( \sqrt{5} \) is an irrational number, \( 56\sqrt{5} \) is also irrational.
After analyzing all options, the only choice that yields an irrational number is: D. \( 4\sqrt{5} \cdot 2\sqrt{49} \).
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