5) Выполните действия: 1) \((2-8i)(1+3i)+13i\); 2) \(1-3i(11i^2+14i^3)-7i^4\); 3) \(i^4+21+\frac{7+i}{5+6i}\); 4) \(20(i+9)(i^4-5)\); 5) \(3i+\frac{1-7i}{4+i}\).

1. \((2-8i)(1+3i)+13i=26-2i+13i=26+11i\).
2. Так как \(i^2=-1\), \(i^3=-i\), \(i^4=1\), получаем: \(1-3i(11i^2+14i^3)-7i^4=1-3i(-11-14i)-7=-48+33i\).
3. \(i^4+21+\frac{7+i}{5+6i}=22+\frac{(7+i)(5-6i)}{61}=22+\frac{41-37i}{61}=\frac{1383}{61}-\frac{37}{61}i\).
4. \(20(i+9)(i^4-5)=20(i+9)(1-5)=-80(i+9)=-720-80i\).
5. \(3i+\frac{1-7i}{4+i}=3i+\frac{(1-7i)(4-i)}{17}=3i+\frac{-3-29i}{17}=-\frac3{17}+\frac{22}{17}i\).

Ответ: 1) \(26+11i\); 2) \(-48+33i\); 3) \(\frac{1383}{61}-\frac{37}{61}i\); 4) \(-720-80i\); 5) \(-\frac3{17}+\frac{22}{17}i\).