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Potassium hydroxide, KOH, conducts electricity because it is an electrolyte. An electrolyte is a compound that dissociates into it's respective ions. An ion is an atom or molecule that has gained or lost electrons, thus giving it a charge. When KOH is dropped into water, it dissociates (breaks down) into it's ions, K+ and OH-.

The positive K+ ion has lost an electron and is looking for another one. The negative OH- ion has an extra electron to give. This constant transfer of electrons is what allows KOH to conduct electricity.

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