Explanation:
remember what an exponent represents : the number of times the base is multiplied by itself. e.g.
a⁵ = a×a×a×a×a
therefore, as example for our case here :
0.71⁴ = 0.71×0.71×0.71×0.71
so, now, what happens, when a number is multiplied by a positive number smaller than 1 ? is the result bigger or smaller than the original number ?
in other words : is 0.71×0.71 bigger or smaller than 0.71 ?
it is smaller, of course.
if you multiply by a number larger than 1, the result will be bigger. if you multiply by a number smaller than 1, the result will be smaller.
so, now back to our given problem here :
P(t) = 420(0.71)^t
the initial population we get for t = 0 :
P(0) = 420(0.71)⁰ = 420×1 = 420
remember, a⁰ = 1 for every a.
the function represents decay, because with every additional year we multiply by 0.71 an additional time, which lowers the result as explained.
by what % did the population change each year ?
it changes by the factor 0.71.
a factor of 1 stands for 100%, meaning the population would stay at 100% of the last year.
the factor 0.71 means that the populating goes down to 71% compared to the 100% of the previous year. so, it goes down (changes) by 100-71 = 29% each year.