1. **State the problem:** Find the distance between the two points $(-6, 8)$ and $(9, -4)$ on the Cartesian plane. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem, where the difference in $x$ and $y$ coordinates form the legs of a right triangle. 3. **Calculate the differences:** $$x_2 - x_1 = 9 - (-6) = 9 + 6 = 15$$ $$y_2 - y_1 = -4 - 8 = -12$$ 4. **Substitute into the formula:** $$d = \sqrt{15^2 + (-12)^2} = \sqrt{225 + 144} = \sqrt{369}$$ 5. **Simplify the radical:** Factor 369: $$369 = 9 \times 41$$ Since $9$ is a perfect square: $$d = \sqrt{9 \times 41} = \sqrt{9} \times \sqrt{41} = 3\sqrt{41}$$ 6. **Final answer:** The distance between the points is $$\boxed{3\sqrt{41}}$$ This is the simplest radical form of the distance.