The half-life of cobalt—60 is 5.27 years. How many milligrams of cobalt-60 remain after 52.7 years if you start with 10.0 mg?
The important thing to recognize here is that the amount of time that passes is a whole number multiple of the isotope's nuclear half-life.
As you know, the equation for nuclear half-life calculations looks like this
#color(blue)(A = A_0 * 1/2^n)" "#, where
In your case, you know that you're interested in finding out how much cobalt-60 would be left undecayed after
#n = (52.7 color(red)(cancel(color(black)("years"))))/(5.27 color(red)(cancel(color(black)("years")))) = 10#
This means that you have
#A = A_0 * 1/2^(10) = A_0/1024#
Plug in the value you have for the initial mass of the sample to get
#A = "10.0 mg"/1024 = color(green)("0.00977 mg") ->#rounded to three sig figs