Both brought up hard-fought half-centuries and, with dark storm clouds looming, kept the scoreboard ticking over at a fast clip.
“We’re in very good position, obviously three down at the end of the end of play,” said Brook.
“Hopefully we can make the most of that going into tomorrow.”
England came into the game buoyed by a four-wicket win inside two days in the previous test in Melbourne, desperate to keep the momentum going.
That victory snapped a 15-year winless streak in Australia but came too late to save the series, with the hosts retaining the urn by winning in Perth, Brisbane and Adelaide.
The day began with a tribute to first responders at the Bondi mass shooting last month that killed 15 people, with huge cheers when hero Ahmed Al Ahmed, who tackled one of the gunmen, appeared.
Duckett was lively when play started, crunching five boundaries from Mitchell Starc in a quickfire 27 off 24 balls.
But England’s tormentor-in-chief Mitchell Starc had the last laugh, enticing an outside edge from an angled ball to wicketkeeper Alex Carey at full stretch.
It was the fifth time Starc has bagged Duckett this series.
Crawley was next to go, trapped lbw by Michael Neser, with both openers back in the sheds by the first drinks break and the tourists in trouble on 51-2.
A cautious Bethell took 15 balls to get off the mark and never looked confident.
He departed after prodding at a moving delivery from Scott Boland that took a faint edge to Carey as England fell to 57-3.
Root joined Brook at the crease and they began to rebuild.
They rotated the strike well and punished any loose balls with Root bringing up his 67th half-century, and the 100-partnership, with a single off Webster.
Only Indian great Sachin Tendulkar, with 68, has scored more test fifties.
Brook was fortunate to survive on 45 when he slogged Starc and the ball dropped between three chasing fielders.
But he kept his composure to reach a 15th half-century four balls after Root, cracking Webster through the covers for a boundary.
AFP
