To find the exact distance $AC$ between the points $A(2, -2)$ and $C(3, -4)$, we can use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the coordinates of points $A$ and $C$:

$AC = \sqrt{(3 - 2)^2 + (-4 + 2)^2}$

Let's calculate this step by step:

1. $(3 - 2) = 1$
2. $(-4 + 2) = -2$

Now square both values:

3. $1^2 = 1$
4. $(-2)^2 = 4$

Add the squares:

$1 + 4 = 5$

Finally, take the square root:

$AC = \sqrt{5}$

So, the exact distance $AC = \sqrt{5}$.

Would you like to know more details or have any questions about this process?

Here are five related questions for further exploration:

1. How would you calculate the distance between points $B(4, -6)$ and $C(3, -4)$?
2. What is the distance between points $A(2, -2)$ and $B(4, -6)$?
3. Can you find the midpoint of segment $AC$?
4. How would the distance formula change if the points were in three-dimensional space?
5. What would the distance formula look like in a non-Euclidean geometry?

Tip: Always double-check the signs of the coordinates when using the distance formula to avoid common mistakes.