Smooth is both an adjective and a verb. If you want to make something smooth, you smooth it. Some dictionaries list smoothen, a verb meaning to make or become smooth, but the word is superfluous and can always give way to smooth.
Smoothe, which appears about a tenth as often as smoothen, is an old secondary spelling of smooth, but it does not commonly appear in 21st-century published writing.
In each of these sentences, smoothe or smoothen could give way to the more conventional smooth:
Impress Skin Care is a skincare routine with clinically proven ingredients to lift, tighten, smoothe, coloration & brighten skin. [AZ Family]
Blue Line trains will share a track between Stadium-Armory and Addison Road-Seat Pleasant while repairs are made to slabs under the rails to smoothen the ride. [Washington Post]
He and Mr Maroni met last week to smoothe over their row over the handling of migrants … [The Economist]
[L]ong-term investors should make only a small allocation, of 5% or less, to oil or oil-related securities as a way to smoothen volatility in their portfolios. [Wall Street Journal]
Although here we’ve used examples from American and British publications, smoothen is especially common in Indian-English publications. We don’t know why this is.