Exterminatory is an uncommon word that means "relating to or tending to destruction or extermination." There are a few similar words that could be used instead depending on the context. Destructive, ruinous, devastating, catastrophic, apocalyptic, and fatalistic all have connotations of great harm or destruction. Omnicidal could also be used in a more extreme sense, meaning a desire to destroy all life on Earth. If the context is more specific, such as in relation to pests, then words like pestilent, blight, and infestation could be used instead. Ultimately, the choice of synonym will depend on the specific situation and the desired meaning.