It is imbibed that we wash ourselves with soap well, especially after playing in the dirt, which gives rise to the impression that soap is that magic substance removing all the 'bad things' from us. Well, that's only half the truth. Soap works by the action of micelles, but more practically speaking, scrubbing is essential - to increase contact surface area by producing more lather - so if your hands are really dirty, the soap makes it easier for the dirt to be carried away by it from the dirty surface, but it in itself is not capable of removing anything automatically. So simply contacting your hand with soap isn't going help. This applies to vessels and any other surface which we clean with soap - clothes, cars, the floor, windows and the kitchen countertop.
One fine sunday a decade or so ago, I decided I hadn't brushed my teeth well enough and had to 'make up' for it by brushing for longer and harder. I didn't repeat it afterward, thankfully, and now realise it was but a futile exercise. That's not how toothpaste and a toothbrush work. Toothpaste in itself is a form of chalk - albeit finely ground chalk, with the toothbrush acting as the tool to use the toothpaste. When we 'brush', we use the abrasive power of the toothpaste to remove the plaque collected on our teeth with the toothbrush additionally removing particles of food lodged in between the teeth.
If you take some toothpaste and grind it between your incisors, you can feel the solid particles. It's something similar to using sandpaper to polish metal or wood. But, beware, unlike those substances which have more if you remove the top layer, teeth are pretty sensitive, and excessive brushing can damage the enamel, i.e. the outermost layer of the teeth.
On a similar note, a full strip of toothpaste on your toothpaste isn't needed - it's just a marketing gimmick used in ads to get you to use as much as the model in the ad is using. Just about a third of the full strip is sufficient. This gimmick extends well to shampoo and laundry detergent also, to the extent it does more damage than good.
Talking of which, there is one substance which works like magic - deicing fluid and rock salt-water for snow. Based on the principle of depression in freezing point, it is very effective at removing ice and snow at not-so low temperatures. But at the same time, it is essential that the salt is added as a solution - it is very common for people to put salt on snow and expect it to disappear - add water and it will.