Which word means to cut down with an ax?

The word that means to cut down with an ax is "hew." In this context, "hew" specifically refers to the act of chopping or cutting something, particularly wood or timber, using a tool like an ax. It conveys a sense of physical labor and precision in shaping or reducing large pieces of material, often used in forestry or carpentry.

The other options do not relate to the act of cutting with an ax. "Imbroglio" refers to a complicated or intricate situation, "hubris" relates to excessive pride or self-confidence, and "idyll" describes a peaceful or picturesque scene, often of rural life. These meanings clearly distinguish them from the definition of "hew," thus confirming that "hew" is the correct term for cutting down with an ax.