1. The problem asks to find the distance between two points $(1,1)$ and $(-7,-8)$ on the Cartesian plane. 2. We use the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ 3. Substitute the given points: $$d=\sqrt{(-7-1)^2+(-8-1)^2}$$ 4. Simplify inside the parentheses: $$d=\sqrt{(-8)^2+(-9)^2}$$ 5. Calculate the squares: $$d=\sqrt{64+81}$$ 6. Add the values: $$d=\sqrt{145}$$ 7. The simplest radical form of the distance is $$\boxed{\sqrt{145}}$$. This means the distance between the points is the square root of 145 units.