We are given the expression $\frac{1}{\sqrt{27}}$ and need to express it in its simplest form with a rational denominator.

Steps:

1. Simplify the square root:

   • $\sqrt{27}$ can be simplified because 27 is $9 \times 3$, and $\sqrt{9} = 3$.
   • So, $\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$.

2. Rewrite the expression:

   • The expression becomes $\frac{1}{3\sqrt{3}}$.

3. Rationalize the denominator:

   • To eliminate the square root from the denominator, multiply both the numerator and the denominator by $\sqrt{3}$.
   • $\frac{1}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3 \times 3} = \frac{\sqrt{3}}{9}$.

Thus, the simplest form of $\frac{1}{\sqrt{27}}$ with a rational denominator is $\frac{\sqrt{3}}{9}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify other square roots like $\sqrt{75}$?
2. What is the process for rationalizing denominators with larger square roots?
3. How do you handle cube roots in denominators?
4. What is the general formula for rationalizing denominators with higher roots?
5. What are some tips for recognizing perfect squares in expressions?

Tip: Always check if the square root can be simplified by factoring it into perfect squares.