The idea here is that the amount of radium-226 you have in your sample will be halved with every passing half-life.
This is the case because a radioactive nuclide's nuclear half-life tells you the amount of time needed for half of an initial sample to undergo radioactive decay.
Mathematically, you can write this as
#A_t = A_0 * (1/2)^color(red)(n)#
#A_t#is the amount of the radioactive nuclide that remains undecayed after a period of time #t#
#A_0#is the initial amount of the radioactive nuclide
#color(red)(n)#is the number of half-lives that pass in the period of time #t#
Now, you know that
#A_t = A_0 * 1/8#
Since you know that
#8 = 2 * 2 * 2 = 2^3#
you can rewrite this as
#A_t = A_0 * (1/2)^color(red)(3)#
You can thus say that in order for the initial sample to be reduced to
Since you know that radium-226 has a half-life of
#3 color(red)(cancel(color(black)("half-lives"))) * "1599 years"/(1color(red)(cancel(color(black)("half-life")))) = color(darkgreen)(ul(color(black)("4797 years")))#
The answer is rounded to four sig figs, the number of sig figs you have for the half-life of the nuclide.
Keep in mind that you have three sig figs for the initial mass of the sample, but you're not using this value in your calculations.