Question

express in simplest form with a rational denominator.\\(\dfrac{5}{\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, \(\frac{5}{\sqrt{63}}=\frac{5}{3\sqrt{7}}\).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{7}\) (because the denominator has \(\sqrt{7}\), and multiplying by \(\sqrt{7}\) will make the denominator a rational number).

\[

$$\begin{align*} \frac{5}{3\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}}&=\frac{5\times\sqrt{7}}{3\times\sqrt{7}\times\sqrt{7}}\\ &=\frac{5\sqrt{7}}{3\times7}\\ &=\frac{5\sqrt{7}}{21} \end{align*}$$

\]

Answer:

\(\frac{5\sqrt{7}}{21}\)