Question

express in simplest form with a rational denominator.
\\(\frac{6}{sqrt{90}}\\)

Explanation:

Step 1: 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 fraction becomes \(\frac{6}{3\sqrt{10}}\).

Step 2: Simplify the fraction by canceling common factors

In the fraction \(\frac{6}{3\sqrt{10}}\), the numerator 6 and the denominator's coefficient 3 have a common factor of 3. Dividing 6 by 3 gives 2, and dividing 3 by 3 gives 1. So now the fraction is \(\frac{2}{\sqrt{10}}\).

Step 3: Rationalize the denominator

To rationalize the denominator (make the denominator a rational number), we multiply both the numerator and the denominator by \(\sqrt{10}\) (because \(\sqrt{10}\times\sqrt{10}=10\), which is rational). So we have:

$$ \frac{2\times\sqrt{10}}{\sqrt{10}\times\sqrt{10}}=\frac{2\sqrt{10}}{10} $$

Step 4: Simplify the resulting fraction

Now, we can simplify \(\frac{2\sqrt{10}}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing 2 by 2 gives 1, and dividing 10 by 2 gives 5. So the simplified form is \(\frac{\sqrt{10}}{5}\).

Answer:

\(\frac{\sqrt{10}}{5}\)