Question

1. \\(\frac{5}{6\sqrt{27}}\\)

Explanation:

Step1: Simplify the square root

First, we simplify \(\sqrt{27}\). We know that \(27 = 9\times3\), and \(\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}\) (since \(\sqrt{9} = 3\)). So the expression becomes \(\frac{5}{6\times3\sqrt{3}}\).

Step2: Multiply the denominator terms

Multiply \(6\) and \(3\) in the denominator. \(6\times3 = 18\), so now the expression is \(\frac{5}{18\sqrt{3}}\).

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{3}\).
\[

$$\begin{align*} \frac{5\times\sqrt{3}}{18\sqrt{3}\times\sqrt{3}}&=\frac{5\sqrt{3}}{18\times3}\\ &=\frac{5\sqrt{3}}{54} \end{align*}$$

\]
(We know that \(\sqrt{3}\times\sqrt{3}=3\), so we multiply \(18\) by \(3\) in the denominator.)

Answer:

\(\dfrac{5\sqrt{3}}{54}\)