Question

express in simplest form with a rational denominator.
\\(\dfrac{3}{\sqrt{63}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, factorize 63. We know that \(63 = 9\times7\), and \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)). So, \(\sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7} = 3\sqrt{7}\). Now the expression becomes \(\frac{3}{3\sqrt{7}}\).

Step2: Cancel out common factors

In the fraction \(\frac{3}{3\sqrt{7}}\), the numerator and the denominator have a common factor of 3. Canceling out the 3s, we get \(\frac{1}{\sqrt{7}}\).

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{7}\). So, \(\frac{1\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}=\frac{\sqrt{7}}{7}\) (since \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)).

Answer:

\(\frac{\sqrt{7}}{7}\)