This has me so confused on how you get the answers

The population at the end of the year is
  (year-end population) = (beginning population) + 6.5%×(beginning population)
  = (beginning population)×(1 + 0.065)

That same multiplier applies each year for 9 years. We know that we can use an exponent to describe repeated multiplication, so the population after 9 years will be ...
  (ending population) = (beginning population)×(1.065)⁹

After 9 years, the population will be 350,000×1.065⁹ ≈ 616,900.

To the risk of sounding redundant.