Katy Perry -- You're Gonna Hear Me Roar ... I FINALLY SOLD MY DIVORCE HOUSE

Katy Perry has finally divorced the house she shared with Russell Brand ... and this time she got taken to the cleaners.

Katy bought the pad in 2011 for $6.5 million -- listed it in early 2013 asking around $8 mil but there were no takers.

Fast forward to December 31 ... our sources say the house was finally sold for $5.65.

After brokers fees ... Katy lost more than a million bucks.

The guy who bought it plans to gut the 8,800 sq. foot Hollywood Hills home.