Question

express in simplest form with a rational denominator.
\\(\dfrac{3}{\sqrt{45}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{45}$. We can factor 45 as $9\times5$, and since $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (for $a\geq0, b\geq0$), we have $\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}$. So the fraction becomes $\frac{3}{3\sqrt{5}}$.

Step2: Cancel common factors in numerator and denominator

We can cancel the common factor of 3 in the numerator and denominator. $\frac{3}{3\sqrt{5}}=\frac{1}{\sqrt{5}}$.

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{5}$. $\frac{1}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{5}$.

Answer:

$\frac{\sqrt{5}}{5}$