Question

express in simplest form with a rational denominator.
\\(\dfrac{4}{\sqrt{18}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{18}$. We know that $18 = 9\times2$, and $\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}$. So the expression becomes $\frac{4}{3\sqrt{2}}$.

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{2}$. So we have $\frac{4\times\sqrt{2}}{3\sqrt{2}\times\sqrt{2}}$.

Step3: Simplify the denominator

Simplify the denominator: $3\sqrt{2}\times\sqrt{2}=3\times(\sqrt{2})^2 = 3\times2 = 6$. And the numerator is $4\sqrt{2}$. So the expression is now $\frac{4\sqrt{2}}{6}$.

Step4: Simplify the fraction

We can simplify $\frac{4\sqrt{2}}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So $\frac{4\sqrt{2}\div2}{6\div2}=\frac{2\sqrt{2}}{3}$.

Answer:

$\frac{2\sqrt{2}}{3}$