England, who at 3-0 up had already won the five-match series, were 97 for one at lunch.
Root was 52 not out, with the only other occasion the 22-year-old Yorkshireman had past fifty this series coming when he made his Test-best 180 in England's crushing 347-run win in the second match at Lord's.
Jonathan Trott was unbeaten on nine.
England, however, were 395 runs behind Australia's first innings 462 for nine declared, a total built on Shane Watson's Test-best 176 and Steven Smith's 138 not out -- his maiden century at this level.
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On a sunny day promising good overhead conditions for batting on a sound pitch, England made a solid start to their morning's work in a match where victory would see them win four Tests in a home Ashes series for the first time.
Root clipped Ryan Harris off his legs for four while left-hander Cook square-drove the fast bowler for a boundary.
Peter Siddle then saw a good length ball stylish cover-driven to the boundary by Root.