To find which option, when multiplied by \( \frac{5\sqrt{27}}{1} \), yields a rational number, we need to simplify \( 5\sqrt{27} \):
\[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \]
So:
\[ 5\sqrt{27} = 5 \times 3\sqrt{3} = 15\sqrt{3} \]
Now, we need to see which of the options can be multiplied with \( 15\sqrt{3} \) to produce a rational number.
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Option 1: \( \sqrt{12} \)
\(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)
\(15\sqrt{3} \cdot 2\sqrt{3} = 30\cdot 3 = 90\) (which is rational) -
Option 2: \( \sqrt{9} \)
\(\sqrt{9} = 3\)
\(15\sqrt{3} \cdot 3 = 45\sqrt{3}\) (which is not a rational number) -
Option 3: \( 4\sqrt{2} \)
\(15\sqrt{3} \cdot 4\sqrt{2} = 60\sqrt{6}\) (which is not a rational number) -
Option 4: \( \sqrt{15} \)
\(15\sqrt{3} \cdot \sqrt{15} = 15\sqrt{45} = 15 \cdot 3\sqrt{5} = 45\sqrt{5}\) (which is not a rational number)
From this evaluation, the only option that results in a rational number is \( \sqrt{12} \).