Question

express in simplest form with a rational denominator.\\(\frac{1}{\sqrt{90}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, factorize 90. We know that \(90 = 9\times10\), and \(\sqrt{9\times10}=\sqrt{9}\times\sqrt{10}\) (by the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)). Since \(\sqrt{9} = 3\), we have \(\sqrt{90}=3\sqrt{10}\). So the expression becomes \(\frac{1}{3\sqrt{10}}\).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{10}\) (because the denominator has \(\sqrt{10}\), and multiplying by \(\sqrt{10}\) will make the denominator a rational number). So we have:
\[
\frac{1\times\sqrt{10}}{3\sqrt{10}\times\sqrt{10}}
\]
Simplify the denominator: \(\sqrt{10}\times\sqrt{10}=10\) (by the property \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)). So the expression becomes \(\frac{\sqrt{10}}{3\times10}=\frac{\sqrt{10}}{30}\).

Answer:

\(\frac{\sqrt{10}}{30}\)