To find the number of bugs after 8 days using the equation \( A(t) = 6 \cdot 2^{0.25t} \), we substitute \( t = 8 \):
\[ A(8) = 6 \cdot 2^{0.25 \cdot 8} \]
First, we calculate \( 0.25 \cdot 8 \):
\[ 0.25 \cdot 8 = 2 \]
Now we substitute this back into the equation:
\[ A(8) = 6 \cdot 2^2 \]
Next, we calculate \( 2^2 \):
\[ 2^2 = 4 \]
Now, substituting \( 4 \) back into the equation gives:
\[ A(8) = 6 \cdot 4 = 24 \]
Therefore, the number of bugs after 8 days is \( \boxed{24} \).