Question

express in simplest form with a rational denominator.
\\(\dfrac{4}{\sqrt{24}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{24}$. We can factor 24 as $4\times6$, and since $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (for $a,b\geq0$), we have $\sqrt{24}=\sqrt{4\times6}=\sqrt{4}\times\sqrt{6}=2\sqrt{6}$. So the expression becomes $\frac{4}{2\sqrt{6}}$.

Step2: Simplify the fraction

We can simplify $\frac{4}{2\sqrt{6}}$ by dividing the numerator and the denominator by 2. So $\frac{4\div2}{2\sqrt{6}\div2}=\frac{2}{\sqrt{6}}$.

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{6}$. So $\frac{2\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{2\sqrt{6}}{6}$.

Step4: Simplify the final fraction

We can simplify $\frac{2\sqrt{6}}{6}$ by dividing the numerator and the denominator by 2. So $\frac{2\sqrt{6}\div2}{6\div2}=\frac{\sqrt{6}}{3}$.

Answer:

$\frac{\sqrt{6}}{3}$