Nadal, 17 times a Grand Slam victor, ruthlessly put an end to Serbian qualifier Filip Krajinovic's run at the tournament, converting three of his four break point opportunities to advance 6-3 6-4.
Nadal is a very accomplished player and he has won the Indian Wells Open 3 times in the past.
In match that resumed after rain stopped play just one game in on Monday night, Kohlschreiber said the warmer daytime conditions suited him better, and Djokovic agreed. I think he probably struggled throughout a little bit.
"He's been around for so many years and he doesn't really get overwhelmed by the occasion of playing on center court", he told reporters.
Federer's last-16 opponent is Kyle Edmund, the Brit cruising past Radu Albot 6-3 6-3. Conditions are tough with the glare, and the jump of the ball is sometimes hard to find the rhythm and timing.
Nadal will take on either eighth-seeded American John Isner or 12th-seeded Russian Karen Khachanov in the quarter-finals, trying to stay on track for a possible blockbuster semi-final with five-time Indian Wells champion Roger Federer.More news: Google Releases Android Q Beta for Developers, Early Adopters
More news: "Bomb cyclone" headed our way, and here's what it means
More news: Twitter reacts to Anthony Barr backing out of agreement with Jets
At the other end of the age spectrum, 40-year-old Croatian Ivo Karlovic's run came to an end with a 6-4, 6-3 loss to 25-year-old seventh seed Dominic Thiem. "I made two breaks in the right moment I would say". After stunning Kei Nishikori for a second time in three weeks, the Pole bounced Denis Shapovalov on Wednesday in three sets 7-6 (3), 2-6, 6-3.
"I congratulate him for playing tactically really well and getting me out of my comfort zone", Djokovic said. He beat Jerzy Janowicz 6-4, 7-6 (2) in the 2013 ATP Masters 1000 Rome quarter-finals.
"I think I played a very solid match".
"I felt good from the beginning", said Federer, who is coming off his 100th career title at Dubai last month.
Kecmanovic, ranked 130th in the world, had taken the first set against Yoshihito Nishioka 6-4 when the 74th-ranked Japanese retired because of injury.
The 130rd-ranked Kecmanovic only got a spot in the draw when Kevin Anderson withdrew.