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The win lifted SRH to second place with 17 points from 14 matches, while PBKS finished in ninth spot with 10 points.
RR suffered their fourth loss on the trot.
PBKS skipper Sam Curran hit five fours and three sixes in his match-winning 41-ball unbeaten 63.
Cricket South Africa said that the injury is unlikely to impact Rabada's preparation for the T20 World Cup.
Livingstone's injury is not of serious nature but the England management, as per ESPNCricinfo, decided to give him more time to get it treated prior to their home T20I series against Pakistan, beginning on May 22.
Chasing a stiff target of 242 to stay alive in the playoffs race, PBKS were bowled out for 181 in 17 overs.
Star batter Virat Kohli slammed a 47-ball 92 as RCB posted 241/7 after being invited to bat first. In reply, PBKS were all out for 181 in 17 overs.
The win ended CSK's five-match losing streak at the hands of Punjab Kings.
PBKS spinners Harpreet Brar and Rahul Chahar spun a web around the CSK batters.
PBKS chased down the target of 163 with 13 balls to spare and notched up their fifth win on the trot against the defenidng champions.
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