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The foam that you see after rubbing soap in your hands is a large collection of tiny soap bubbles. Each individual bubble is actually an extremely thin film of the soap solution. Since soap is a surfactant (meaning that it affects the surface tension of the liquid), it lowers the surface tension of water, which then causes the film to stretch. This results in an increase in the total surface area of the film (thanks to the countless bubbles), which causes the already deficient color in the film to spread out and become next to impossible to visually detect.
The foam in its entirety looks white because when light enters the soap solution, it must pass through a number of tiny soap bubbles, i.e., a number of surfaces. These countless surfaces scatter the light in different directions, which makes the foam/lather look white.
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