This is the 9230th most frequent Japanese word.
生き抜い
"生き抜い" is not a standard Japanese word, but "生き抜く" means "to survive" or "to live through (difficult circumstances)."
Here, 生き抜いた is used to express enduring and surviving through hardships in life.
彼は困難な時代を生き抜いた。
He survived through difficult times.
生き抜く is used to describe the ability or power to live through and overcome challenges.
彼女の生き抜く力に感動した。
I was impressed by her strength to survive.
Here, 生き抜こう is a volitional form encouraging action to survive with determination.