You will be left with
A radioactive isotope's nuclear half-life tells you how much time must pass before the half of the original sample undergoes decay.
In your case, the half-life of iodine-131 is 8 days. This means that any sample of iodine-131 that undergoes decay will be halved after 8 days, halved again after another 8 days, halved again after another 8 days, and so on.
You know that the fuel rods are stored underwater for 56 days. You can use this value to determine how many half-lives will pass.
Plug in your values to get
So, if the initial amount gets halved with every passing half-life, you can write
In your case, you have
This means that the fraction remaining after 56 days will be
The first thing to do is count how many half lives have elapsed.
The half life (
You can see that 56 days is 7 half lives since 56/8 = 7
To get the fraction remaining we need to x by 1/2 seven times:
1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/128
If you don't get nice numbers like this you can use the equation for radioactive decay:
We get the decay constant from the half life:
Taking natural logs
To compare this with the other method we can check the value of 1/128 = 0.00781 so that agrees quite nicely.