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Root was on 178, having brought up his 17th test century on Sunday, while Pope had made 46 as the pair took their side's overnight score of 269-5 through to a four-run lead over the hosts' first innings of 375.
It was Root's second partnership in excess of 100 runs in the innings after he combined with Rory Burns (101) in a 177-run third wicket stand that resurrected England from 24-2.
The tourists, however, will likely have to accelerate their scoring rate after the break if they want to try and force a result and level the two-match series.
New Zealand won the first test at Bay Oval in Mt Maunganui by an innings and 65-runs but heavy rain is forecast for Hamilton on the fifth day on Tuesday and, while there has been some uneven bounce, the pitch is still proving easy to bat on.
Root was under pressure entering the match, having suffered a lean run of form since he became captain almost three years ago.
The 28-year-old had also not achieved three figures in a te