Question

express in simplest form with a rational denominator.\\(\dfrac{6}{\sqrt{63}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, factorize 63. We know that \(63 = 9\times7\), and \(\sqrt{9\times7}=\sqrt{9}\times\sqrt{7}=3\sqrt{7}\) (since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a\geq0,b\geq0\)). So the expression becomes \(\frac{6}{3\sqrt{7}}\).

Step2: Simplify the fraction by dividing numerator and denominator by 3

Dividing 6 by 3 gives 2, and dividing \(3\sqrt{7}\) by 3 gives \(\sqrt{7}\). So now the expression is \(\frac{2}{\sqrt{7}}\).

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{7}\). So we have \(\frac{2\times\sqrt{7}}{\sqrt{7}\times\sqrt{7}}\). Since \(\sqrt{7}\times\sqrt{7}=7\) (by the property \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)), the expression simplifies to \(\frac{2\sqrt{7}}{7}\).

Answer:

\(\frac{2\sqrt{7}}{7}\)