the population in my state is 4,413,457 and the Annual Growth Rate is 0.09% what will the population be in ten years

OK What you do here is change the percent to a decimal then add 1. This is 1.0009. Then raise that to the power of 10 like this (1.0009)¹⁰ then multiply by the population 4,413,457.

(1.0009)¹⁰ * 4,413,457.

$${\mathtt{4\,413\,457}}{\mathtt{\,\times\,}}{\left({\mathtt{1.000\: \!9}}\right)}^{{\mathtt{10}}} = {\mathtt{4\,453\,339.370\: \!205\: \!616\: \!109\: \!484\: \!1}}$$

Then round to a whole number 4,453,339.

I look it up your state is Kentucky. 

OK I look here to find the population

https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_population

I compare the 2000 population with the 2010 population for a 10 year average and 2015 with 2000 as an average for 15 years. That give about 0.72% which is 0.0072. The way they do the math it might be higher like 0.009. That is 0.9%

SO this means maybe you meant the number to be 0.9% not 0.09%. 0.9% is 0.009.

If that is true then the answer is

$${\mathtt{4\,413\,457}}{\mathtt{\,\times\,}}{\left({\mathtt{1.009}}\right)}^{{\mathtt{10}}} = {\mathtt{4\,827\,147.417\: \!057\: \!420\: \!231\: \!152\: \!8}}$$

I dont know for sure if it is right but it work for the listed populations so this is probably the right answer, but one of the mods should check though.