Question

The product of 5 v|/27 and blank would yield a rational number
Square root 12
Square root 9
4 square root 2
Square root 15

Answer 1

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.

1. **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)

2. **Option 2: \( \sqrt{9} \)**
\(\sqrt{9} = 3\)
\(15\sqrt{3} \cdot 3 = 45\sqrt{3}\) (which is not a rational number)

3. **Option 3: \( 4\sqrt{2} \)**
\(15\sqrt{3} \cdot 4\sqrt{2} = 60\sqrt{6}\) (which is not a rational number)

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