Question

the product of 5 radical 27 and what would yield a rational number

Answer 1

To find a number that, when multiplied by \( 5 \sqrt{27} \), results in a rational number, we can start by simplifying \( 5 \sqrt{27} \).

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

Now, we need to find a number \( x \) such that:
\[
15 \sqrt{3} \cdot x \text{ is rational}.
\]
To eliminate the irrational part, we can multiply \( 15 \sqrt{3} \) by \( \sqrt{3} \):
\[
15 \sqrt{3} \cdot \sqrt{3} = 15 \cdot 3 = 45,
\]
which is indeed a rational number.

Therefore, the value of \( x \) that would yield a rational number when multiplied by \( 5 \sqrt{27} \) is:
\[
\sqrt{3}.
\]