Old English sape "soap, salve" (originally a reddish hair dye used by Germanic warriors to give a frightening appearance), from Proto-Germanic *saipon "dripping thing, resin" (source also of Middle Low German sepe, West Frisian sjippe, Dutch zeep, Old High German seiffa, German seife "soap," Old High German seifar "foam," Old English sipian "to drip"), from PIE *soi-bon-, from root *seib- "to pour out, drip, trickle" (perhaps also the source also of Latin sebum "tallow, suet, grease").
Romans and Greeks used oil to clean skin; the Romance words for "soap" (Italian sapone, French savon, Spanish jabon) are from Late Latin sapo "pomade for coloring the hair" (first mentioned in Pliny), which is a Germanic loan-word, as is Finnish saippua. The meaning "flattery" is recorded from 1853.
Soap opera is recorded from 1939, as a disparaging reference to daytime radio dramas sponsored by soap manufacturers.
Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing.
The city in Somerset, England (Old English Baðun) was so called from its hot springs. Bath salts is attested from 1875 (Dr. Julius Braun, "On the Curative Effects of Baths and Waters"). Bath-house is from 1705; bath-towel is from 1958.
also soapbox, 1650s, "box for holding soap," later especially a wooden crate in which soap may be packed; from soap (n.) + box (n.). Typical of a makeshift stand for a public orator at least since 1907. Also used by children to make racing carts, as in soap-box derby, annual race in Dayton, Ohio, which dates to 1933.