There are two sentences:
- She is living out at her home.
- She is living out her life at her home.
I was said that the first is wrong as It doesn't contain "her life". Is it really so much important to have this addition If my context is about dying?
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To "live out somewhere" is a completely different verb from to "live your life out". In fact, the two can be combined into a single sentence:
She's living her life out out at her home.
In "live out at her home", "live" is a stand-alone verb with the normal meaning, and "out at her home" is an adverbial that gives the location of living.
In "live out her life", "live out" is a phrasal verb with the object "her life". This phrasal verb means roughly to live the full extent of some time period, usually the person's life, but it can also be things like a prison sentence.