To determine which option would yield a rational number when multiplied by \( 5\sqrt{27} \), we first simplify \( 5\sqrt{27} \).
We know that:
\[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \]
So:
\[ 5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3} \]
Now we need to find which option, when multiplied with \( 15\sqrt{3} \), results in a rational number. A product will be rational if the square root factors cancel out.
Let’s analyze each option:
A. \( \sqrt{12} \)
\[ 15\sqrt{3} \cdot \sqrt{12} = 15\sqrt{3} \cdot \sqrt{4 \cdot 3} = 15\sqrt{3} \cdot (2\sqrt{3}) = 15 \cdot 2 \cdot (\sqrt{3} \cdot \sqrt{3}) = 30 \cdot 3 = 90 \quad (\text{Rational}) \]
B. \( \sqrt{9} \)
\[ 15\sqrt{3} \cdot \sqrt{9} = 15\sqrt{3} \cdot 3 = 45\sqrt{3} \quad (\text{Not Rational}) \]
C. \( 4\sqrt{2} \)
\[ 15\sqrt{3} \cdot 4\sqrt{2} = 60\sqrt{6} \quad (\text{Not Rational}) \]
D. \( \sqrt{15} \)
\[ 15\sqrt{3} \cdot \sqrt{15} = 15\sqrt{45} = 15\cdot 3\sqrt{5} = 45\sqrt{5} \quad (\text{Not Rational}) \]
From these findings, only Option A \( \sqrt{12} \) yields a rational number when multiplied by \( 15\sqrt{3} \).
Thus, the answer is:
A. Square root of 12