Question

The product of 5sqrt27 and BLANK would yield a rational number.
a. sqrt12
b.sqrt9
c.4 sqrt2
d.sqrt15

Answer 1

To determine which option would yield a rational number when multiplied by \(5\sqrt{27}\), we need to find the value that will create a perfect square under the square root when the two values are multiplied.

First, let's simplify \(5\sqrt{27}\):
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]
Thus,
\[
5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3}
\]

Next, let's analyze each option to see which will yield a rational product:

a. \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \)
\[
15\sqrt{3} \times 2\sqrt{3} = 30 \times 3 = 90 \quad \text{(rational)}
\]

b. \( \sqrt{9} = 3 \)
\[
15\sqrt{3} \times 3 = 45\sqrt{3} \quad \text{(not rational)}
\]

c. \( 4\sqrt{2} \)
\[
15\sqrt{3} \times 4\sqrt{2} = 60\sqrt{6} \quad \text{(not rational)}
\]

d. \( \sqrt{15} \)
\[
15\sqrt{3} \times \sqrt{15} = 15\sqrt{45} = 15 \cdot 3\sqrt{5} = 45\sqrt{5} \quad \text{(not rational)}
\]

From the calculations, the only option that yields a rational number when multiplied by \(15\sqrt{3}\) is:

**a. \( \sqrt{12} \)**