To simplify the given expression $\frac{7}{\sqrt{12}}$ and express it with a rational denominator, we need to rationalize the denominator. Here's the step-by-step process:

Step 1: Simplify the square root.

$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ Thus, the expression becomes: $\frac{7}{\sqrt{12}} = \frac{7}{2\sqrt{3}}$

Step 2: Rationalize the denominator.

To get rid of the square root in the denominator, multiply both the numerator and the denominator by $\sqrt{3}$: $\frac{7}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{7\sqrt{3}}{2 \times 3} = \frac{7\sqrt{3}}{6}$

Final answer:

$\frac{7\sqrt{3}}{6}$

So, $\frac{7}{\sqrt{12}}$ simplifies to $\frac{7\sqrt{3}}{6}$.

Would you like further explanation or details on any part of the solution?

Here are five related questions you might explore:

1. How do you simplify a square root involving multiplication?
2. What does it mean to rationalize a denominator, and why is it done?
3. Can you express $\frac{7\sqrt{3}}{6}$ in decimal form?
4. What are other examples where rationalizing the denominator is necessary?
5. How would the process change if the denominator was $\sqrt{18}$ instead?

Tip: Rationalizing the denominator is a common practice in simplifying expressions to make them easier to work with in further calculations.