In a grandiloquent manner; using grand or pompous language and style.
From grandiloquent (the adjective form of grandiloquence) plus -ly (Old English suffix forming adverbs from adjectives). The adverb form allows the quality of being grandiloquent to modify verbs or other parts of speech.
Adding -ly to adjectives to make adverbs is so standard in English that we barely notice it, but it's actually quite recent historically. Old English didn't have this system; the -ly suffix was repurposed from an Old English word meaning 'like' to create our modern adverb system.
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