Step 1: Note the data.
Starting nuclei $N_0=8\times10^{16}$, half-life $T_{1/2}=15$ days, total time $t=60$ days. We want how many have decayed.
Step 2: Count the half-lives.
Number of half-lives is \[ n=\frac{t}{T_{1/2}}=\frac{60}{15}=4. \] So four half-lives pass.
Step 3: Recall the decay rule.
After $n$ half-lives the number still left is \[ N=N_0\left(\frac{1}{2}\right)^n. \]
Step 4: Compute the surviving nuclei.
\[ N=8\times10^{16}\times\left(\frac{1}{2}\right)^4=8\times10^{16}\times\frac{1}{16}=0.5\times10^{16}. \]
Step 5: Understand "decayed".
Decayed nuclei are the ones that are no longer active, i.e. the starting number minus the survivors.
Step 6: Subtract.
\[ N_{\text{decayed}}=8\times10^{16}-0.5\times10^{16}=7.5\times10^{16}. \]
Step 7: State the result.
The number decayed is $7.5\times10^{16}$, which is option (1).
\[ \boxed{7.5\times10^{16}} \]