If we start with 8000 atoms of radium-226, how much would remain after 3,200 years?
The problem doesn't provide you with the nuclear half-life of the radium-226 isotope, so you're going to have to do a bit of research on that.
You can find this isotope's half-life listed as
#t_"1/2" = "1600 years"#
So, the half-life of a radioactive nuclide tells you how much time must pass in order for half of an initial sample of said nuclide to undergo radioactive decay.
In your case, the half-life of radium-226 tells you that a sample of this nuclide will need
Your starting sample contains
#"8000 atoms" * 1/2 = "400 atoms " ->#after #1600#years
How about after two half-lives pass? The remaining sample will be halved again
#"4000 atoms" * 1/2 = "2000 atoms " ->#after #3200#years
Therefore, your initial sample of radium-226 will be down to
#"no. of atoms that remain undecayed" = color(green)(|bar(ul(color(white)(a/a)color(black)("2000 atoms")color(white)(a/a)|)))#