a). If the half-life is 40 days, then 5 half-lives = (5 x 40) days.
b). If 5 half-lives have passed, then the radioactive portion of
the sample has been cut in half five times.
The amount that remains is (1/2)⁵ of the original amount.
That's (the original amount) x (1/2 · 1/2 · 1/2 · 1/2 · 1/2)
or
(the original amount) / 2⁵
or
(the original amount) / 32 .