〜ゆえ(に)・〜がゆえに
because of, therefore (literary)
Noun (の) + ゆえ(に) · Plain form + が + ゆえ(に) · Sentence。ゆえに、…
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ゆえ (故) is a literary "because": 若さゆえに、失敗することもある, "being young, you sometimes fail". After a verb or adjective, it becomes がゆえに: 正直であるがゆえに、損をする, "he loses out precisely because he's honest".
At the start of a sentence, ゆえに means "therefore", as in logic and philosophy: 我思う、ゆえに我あり, "I think, therefore I am".
It's common in essays, speeches, song lyrics and anime, wherever a weighty, old-fashioned tone is wanted. In conversation, you'd use から, ので, or ために.
The kanji 故 is also in 故郷 (hometown) and 事故 (accident), though with different readings.
Examples
若さゆえに、失敗することもある。
Being young, you sometimes fail.
貧しさゆえに、学校に行けない子どもたちがいる。
There are children who can't go to school because of poverty.
彼は正直であるがゆえに、損をすることが多い。
He often loses out precisely because he's honest.
愛するがゆえに、厳しくするのだ。
It's because I love you that I'm strict.
我思う、ゆえに我あり。
I think, therefore I am.