It breathes. Simple as that.
No. A lioness and a polar bear would not meet because one (the lioness) lives in a hot, arid environment and the other (being the polar bear) lives in a very cold dry environment. However, if they did meet, the polar bear may be able to kill the lioness with one single swipe to the head. The lioness, on the other hand, would need other lions and lionesses in order to bring down the polar bear.
a polar bear needs its food sources and fur to keep it warm
A lot.
Polar bears do not live in the rainforest but they do depend upon it for the oxygen it produces.
A polar bear is a mammal and all mammals need air to breath.
A polar bear would need to adjust a lot of things in order to survive in a desert. Without air conditioning, or millions of years to evolve, a polar bear would likely die in the new environment.
Polar bears need to survive with water, food, nutrients, and they are fur to stay warm
polar bears need to live, so they have to get energy from other animals.
No, the cold does not affect polar bear cubs because they have the same percent of body fat as an adult would need.
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Not in the literal sense, but they can really burn up the ice. A typical polar bear can burn along at around 30 miles per hour. Remember: you don't need to be able to out-run the bear...you only need to out-run the people who are with you.