1. The problem is to find the distance between the two points $(3, -3)$ and $(7, -7)$ on the coordinate plane. 2. We use the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem, where the distance is the hypotenuse of a right triangle formed by the horizontal and vertical differences. 3. Substitute the given points into the formula: $$d = \sqrt{(7 - 3)^2 + (-7 - (-3))^2}$$ 4. Simplify inside the parentheses: $$d = \sqrt{4^2 + (-4)^2}$$ 5. Calculate the squares: $$d = \sqrt{16 + 16}$$ 6. Add the values inside the square root: $$d = \sqrt{32}$$ 7. Simplify the radical by factoring out perfect squares: $$d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$$ 8. Therefore, the distance between the points $(3, -3)$ and $(7, -7)$ in simplest radical form is: $$\boxed{4\sqrt{2}}$$