Question

question
express in simplest form with a rational denominator.
\\(\frac{5}{\sqrt{54}}\\)
answer attempt 1 out of 10

Explanation:

Step1: Simplify the square root in the denominator

First, we factorize 54. We know that \(54 = 9\times6\), and \(\sqrt{9\times6}=\sqrt{9}\times\sqrt{6} = 3\sqrt{6}\) (since \(\sqrt{9} = 3\)). So the original expression \(\frac{5}{\sqrt{54}}\) becomes \(\frac{5}{3\sqrt{6}}\).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{6}\) (because the denominator has \(\sqrt{6}\), and multiplying by \(\sqrt{6}\) will make the denominator a rational number).

So we have:

$$ \frac{5}{3\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}=\frac{5\times\sqrt{6}}{3\times(\sqrt{6}\times\sqrt{6})} $$

Step3: Simplify the denominator

We know that \(\sqrt{6}\times\sqrt{6}=(\sqrt{6})^2 = 6\). So the denominator becomes \(3\times6 = 18\), and the numerator is \(5\sqrt{6}\).

So the simplified form with a rational denominator is \(\frac{5\sqrt{6}}{18}\).

Answer:

\(\frac{5\sqrt{6}}{18}\)