Is the word exausto the same in Brazilian and European Portuguese?

Yes

The word "exausto" is identical in Brazilian and European Portuguese regarding its spelling, meaning, and grammatical function (adjective). The differences are exclusively phonetic; Brazilian Portuguese tends to pronounce the vowels more openly and with less reduction, whereas European Portuguese often features more closed or reduced vowels and a distinct rhythmic cadence.

Brazilian Portuguese

1. Estou exausto depois de trabalhar o dia todo. (I am exhausted after working all day.)
2. Ela ficou exausta com a maratona. (She became exhausted with the marathon.)
3. O atleta parecia exausto ao cruzar a linha de chegada. (The athlete looked exhausted upon crossing the finish line.)
4. Eu estou morto de cansaço hoje. (I am dead tired today.)
5. Estamos exaustos com tanta notícia ruim. (We are exhausted with so much bad news.)

European Portuguese

1. Estou estafado depois de trabalhar o dia todo. (I am worn out after working all day.)
2. Ela ficou estafada com a maratona. (She became worn out with the marathon.)
3. O atleta parecia estafado ao cruzar a linha de chegada. (The athlete looked worn out upon crossing the finish line.)
4. Depois da viagem, eu estava mesmo exausto. (After the trip, I was truly exhausted.)
5. Ficámos estafados com o calor intenso. (We became worn out with the intense heat.)