Pour 2 cups of baking soda into the bottom of a clean 1-gallon bucket. Run the hot water in the sink and allow it to get as hot as possible before filling the bucket with hot water.
Stir the baking soda and hot water mixture with a wooden stirrer to fully combine the ingredients. There should be no particles of baking soda left in the bottom of the bucket.
Gently wipe the mixture over the exterior wall, using a large sponge with a soft surface. Do not scrub the wall; just coat it with the baking soda mixture. Allow the baking soda mixture to rest on the surface of the wall for 20 to 30 minutes.
Rub the exterior slate wall with a soft brush or soft scrubbing pad to remove any dirt and debris from the wall. After scrubbing the wall, rinse it with clean, cold water from a hose.
Allow the exterior slate wall to dry naturally before inspecting it for remaining dirt. If the wall is not as clean as you want, repeat the process, allowing the baking soda mixture to rest for 20 to 30 minutes before scrubbing the walls with a soft brush or scrubbing pad.