The product of 5to the square root of 27 and ____ would yield a rational number.

A square root of 12
B square root of 9
C 4 to the square root of 2
D square root of 15

1 answer

To determine which option, when multiplied with \(5 \sqrt{27}\), yields a rational number, we first simplify \(5 \sqrt{27}\):

\[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3} \]

Thus, we have:

\[ 5 \sqrt{27} = 5 \cdot 3 \sqrt{3} = 15 \sqrt{3} \]

Next, we look at each option and check if multiplying it by \(15 \sqrt{3}\) results in a rational number.

Option A: \( \sqrt{12} \)

\[ 15 \sqrt{3} \cdot \sqrt{12} = 15 \sqrt{3} \cdot \sqrt{4 \cdot 3} = 15 \sqrt{3} \cdot 2 \sqrt{3} = 15 \cdot 2 \cdot 3 = 90 \]

This is rational.


Option B: \( \sqrt{9} \)

\[ 15 \sqrt{3} \cdot \sqrt{9} = 15 \sqrt{3} \cdot 3 = 45 \sqrt{3} \]

This is not rational.


Option C: \( 4 \sqrt{2} \)

\[ 15 \sqrt{3} \cdot 4 \sqrt{2} = 60 \sqrt{6} \]

This is not rational.


Option D: \( \sqrt{15} \)

\[ 15 \sqrt{3} \cdot \sqrt{15} = 15 \sqrt{3 \cdot 15} = 15 \sqrt{45} = 15 \sqrt{9 \cdot 5} = 15 \cdot 3 \sqrt{5} = 45 \sqrt{5} \]

This is not rational.


After examining all options, only Option A: \(\sqrt{12}\) results in a rational number when multiplied with \(15 \sqrt{3}\).

So the answer is:

A. \(\sqrt{12}\)