Bài 4.10 trang 201 SBT giải tích 12

Tính các lũy thừa sau:

a) \({\left( {3 - 4i} \right)^2}\)             

b) \({\left( {2\; + 3i} \right)^3}\)

c) \({\left[ {\left( {4\; + {\rm{ }}5i} \right) - \left( {4\; + 3i} \right)} \right]^5}\)

d) \({(\sqrt 2  - i\sqrt 3 )^2}\)

a) \({(3 - 4i)^2} = {3^2} - 2.3.4i + {(4i)^2}\)\( = 9 - 24i - 16 =  - 7 - 24i\)

b) \({(2 + 3i)^3}\)\( = {2^3} + {3.2^2}.3i + 3.2.{(3i)^2} + {(3i)^3}\) \( = 8 + 36i - 54 + 27{i^3}\) \( =  - 46 + 36i - 27i\) \( =  - 46 + 9i\)

c) \({\left[ {(4 + 5i) - (4 + 3i)} \right]^5} = {\left( {2i} \right)^5}\)\( = {2^5}.{i^5} = 32.{i^4}.i = 32i\)

d) \({\left( {\sqrt 2  - i\sqrt 3 } \right)^2}\)\( = 2 - 2.\sqrt 2 .\sqrt 3 i + {\left( {i\sqrt 3 } \right)^2}\) \( = 2 - 2\sqrt 6 i - 3 =  - 1 - 2i\sqrt 6 \).