IPA: /ˌeɪksəˈnɪkli/
KK: /æksɛnɪkli/
In a way that does not involve any other living organisms.
The experiment was conducted axenically to ensure no contamination occurred.
The word 'axenically' is derived from 'axenic' (from Greek 'axenos', meaning without strangers or foreign substances) and the suffix '-ly' (meaning in a manner of). Thus, 'axenically' refers to something done in a manner that is free from contamination or foreign substances.
Think of 'axenic' meaning free from foreign substances, and remember that 'axenically' describes doing something in that pure manner.