Question

express in simplest form with a rational denominator.
\\(\frac{6}{\sqrt{54}}\\)
answer attempt 1 out of 10

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify $\sqrt{54}$. We factor 54 into its prime factors: $54 = 9\times6 = 9\times2\times3$. Since $\sqrt{9}=3$, we can rewrite $\sqrt{54}$ as $\sqrt{9\times6}=3\sqrt{6}$. So the expression becomes $\frac{6}{3\sqrt{6}}$.

Step2: Simplify the fraction

We can simplify $\frac{6}{3\sqrt{6}}$ by dividing 6 by 3, which gives $\frac{2}{\sqrt{6}}$.

Step3: Rationalize the denominator

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

Step4: Simplify the final fraction

We can simplify $\frac{2\sqrt{6}}{6}$ by dividing both the numerator and the denominator by 2. This gives $\frac{\sqrt{6}}{3}$.

Answer:

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