Exegesis:

epoiēsen kratos ‘he has done a mighty deed.’

kratos ‘power,’ ‘might,’ or ‘mighty deed.’ When the former meaning is accepted, epoiēsen is to be rendered ‘he showed, or, displayed’ (cf. Revised Standard Version). In the latter meaning the phrase is to be understood as the rendering of the Hebrew expression ‘to do strength,’ i.e. ‘to do a mighty deed, or, mighty deeds’ (cf. e.g. 1 Sam. 14.48; Ps. 60.12). This is preferable. kratos refers either to a single act of God and then takes up the phrase of v. 49: epoiēsen … megala, or to the series of acts described in the subsequent verses, preferably the latter.

en brachioni autou ‘by means of his arm’; en with dative has here instrumental meaning as a translation of the Hebrew be.

brachiōn ‘arm,’ often as a symbol of God’s active power, cf. Is. 52.10; Ps. 71.18; 77.16. The omission of the article before brachioni is due to Hebrew influence (cf. Moule 117).

dieskorpisen ‘he scattered.’

diaskorpizō ‘to disperse’ i.e. ‘to cause to fly in all directions’ cf. Mt. 26.31.

huperēphanous dianoia kardias autōn ‘the proud in the thought of their heart’; for the omission of the article before dianoia see above.

huperēphanos ‘arrogant,’ ‘presumptuous.’

dianoia (also 10.27) ‘understanding,’ ‘mind,’ here ‘thought,’ ‘(way of) thinking’; since dianoia in the Septuagint is one of the renderings of Hebr. lev (heart), of which kardia is the normal rendering, it follows that both words here represent virtually the same idea. dianoia, in the dative, is limitative and denotes as it were the area or the seat of their arrogance; hence translations like “the proudminded” (An American Translation), ‘the men with a proud heart’ (Bible de Jérusalem). The people to whom the phrase refers, are probably the rulers of v. 52 and the rich of v. 53.

Translation:

He has shown strength with his arm, preferably, ‘he has done (or, he has performed/worked) mighty deeds (or powerful works) with his arm,’ ‘he acts powerfully with his arm.’ The instrumental function which ‘his arm’ has in the sentence may have to be expressed by, ‘he uses (or, causes) his arm to do…,’ or by making it the subject of the clause (Javanese). In some languages ‘hand,’ or a term covering both ‘hand’ and ‘arm,’ can better be used in the required metaphorical sense (e.g. in Sranan Tongo, Vai, and in Tboli respectively). Where a metaphorical rendering is impossible one may have to say something like, ‘he has power to do mighty deeds,’ ‘strong as he is he does mighty deeds.’

He has scattered the proud …, or, ‘he causes the proud … to fly in all directions,’ ‘the proud … fled in confusion through him’ (Balinese).

The proud in the imagination of their hearts, or, “the arrogant of heart and mind” (New English Bible), ‘those who think in their heart that they are great, or, people of bigness’ (South Bolivian Quechua, Kituba), ‘big thoughts in the heart thinking people’ (Ekari); or rendering the phrase “imagination of their hearts” by one term, ‘the proud in heart’ (Marathi), ‘those who have proud thoughts’ (Balinese). In Tboli the qualifying phrase has to be omitted because it would suggest a limitation of meaning; in some other languages, where the normal word or phrase for ‘proud’ contains a reference to the seat of pride, such as ‘heart/mind/thought,’ the qualifying phrase has to be omitted to avoid repetition, e.g. ‘the arrogant hearted men’ (Bahasa Indonesia, similarly in Batak Toba), ‘proud/haughty people (lit. big-memory-men)’ (Sranan Tongo). Sundanese possesses three terms, identical in their first two syllables, and meaning respectively ‘proud because of honoured position’ (used here), ‘proud because of strength,’ ‘proud because of intelligence.’

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.