[tex3]a)\ (1-2i)^5=\\ [(1-2i)^2]^2\cdot(1-2i)=\\ [1-4i+4i^2]^2\cdot(1-2i)=\\ [-3-4i]^2\cdot(1-2i)=\\(9+24i+16i^2)\cdot(1-2i)=\\(-7+24i)\cdot(1-2i)=\\-7+14i+24i-48i^2=\\-7+48+38i=\\41+38i[/tex3]
[tex3]b)\ 16i^5+5i^{10}-(3i)^3[/tex3]
Vamos olhar as potências de [tex3]i[/tex3]
depois voltamos para a expressão inicial.
[tex3]i^3=i^2\cdot i=-i\\\therefore\boxed{i^3=-i}\\i^5=i^4\cdot i=(i^2)^2\cdot i=(-1)^2\cdot i=i\\\therefore\boxed{i^5=i}\\i^{10}=(i^5)^2=i^2=-1\\\therefore\boxed{i^{10}=-1}[/tex3]
Logo:
[tex3]16i^5+5i^{10}-(3i)^3=16i-5-27(-i)=-5+16i+27i=-5+43i[/tex3]
Espero ter ajudado
.
Saudações.