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To hoist someone or something up to a higher point. A noun can be used between "boost" and "up" or after "up." I boosted up my daughter so that she could get a better view of the giraffes at the zoo. Boost me up so I can spy on them through the window! Boost up that box before it falls.
boost someone up
to give someone a helpful lift up to something. She boosted me up so I could get into the window. They boosted up the child for a better view.