We are given a right triangle $ABC$ with sides $AC = 7$ and $BC = 4$, where the angle at $C$ is $\frac{\pi}{2}$ (which means it's a right angle). We are asked to find the exact length of side $AB$.

Since the triangle is a right triangle, we can use the Pythagorean theorem, which states:

$AB^2 = AC^2 + BC^2$

Substituting the given lengths:

$AB^2 = 7^2 + 4^2$ $AB^2 = 49 + 16$ $AB^2 = 65$

Therefore, the length of $AB$ is:

$AB = \sqrt{65}$

So, the exact length of side $AB$ is $\sqrt{65}$.

Would you like further details or clarifications?

Here are 5 related questions for further exploration:

1. How do you calculate the area of triangle $ABC$ given the lengths of two sides?
2. How would the length of $AB$ change if $AC$ or $BC$ were different?
3. What are the trigonometric ratios for the angles $A$ and $B$ in triangle $ABC$?
4. Can you determine the angles $A$ and $B$ using inverse trigonometric functions?
5. What is the perimeter of triangle $ABC$?

Tip: When solving right triangles, the Pythagorean theorem is one of the most powerful tools to connect side lengths.