To express $\frac{4}{\sqrt{2}}$ in simplest form with a rational denominator, you need to rationalize the denominator. Here's how:

1. Multiply both the numerator and denominator by $\sqrt{2}$ to eliminate the square root in the denominator: $\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}$

2. Simplify the fraction: $\frac{4\sqrt{2}}{2} = 2\sqrt{2}$

Thus, the simplest form of $\frac{4}{\sqrt{2}}$ with a rational denominator is:

$2\sqrt{2}$

Let me know if you need further clarification or details.

Here are 5 related questions to deepen your understanding:

1. What are the general steps for rationalizing a denominator?
2. Why is it important to have a rational denominator in mathematics?
3. How would you rationalize a denominator with a more complex square root, such as $\frac{5}{\sqrt{3} + 1}$?
4. What happens if the numerator also contains a square root—how do you simplify it?
5. How does this process relate to simplifying expressions with imaginary numbers?

Tip: Always simplify the numerator and denominator fully after rationalizing for the clearest final answer.