Bài 1 trang 138 SGK Giải tích 12

Thực hiện các phép chia sau:

a) \( \dfrac{2+i}{3-2i}\);          b) \( \dfrac{1+i\sqrt{2}}{2+i\sqrt{3}}\);

c) \( \dfrac{5i}{2-3i}\);            d) \( \dfrac{5-2i}{i}\).

a) \(\dfrac{{2 + i}}{{3 - 2i}} = \dfrac{{\left( {2 + i} \right)\left( {3 + 2i} \right)}}{{\left( {3 - 2i} \right)\left( {3 + 2i} \right)}} \) \(= \dfrac{{6 + 7i + 2{i^2}}}{{9 + 4}} = \dfrac{4}{{13}} + \dfrac{7}{{13}}i.\)

b) \(\dfrac{{1 + i\sqrt 2 }}{{2 + i\sqrt 3 }} = \dfrac{{\left( {1 + i\sqrt 2 } \right)\left( {2 - i\sqrt 3 } \right)}}{{\left( {2 + i\sqrt 3 } \right)\left( {2 - i\sqrt 3 } \right)}}\)

\(= \dfrac{{ 2 + \left( {2\sqrt 2 - \sqrt 3 } \right)i - \sqrt 6 {i^2}}}{{4 + 3}} \) \(= \dfrac{{ 2 + \sqrt 6 }}{7} + \dfrac{{2\sqrt 2 - \sqrt 3 }}{7}i.\)

c) \(\dfrac{{5i}}{{2 - 3i}} = \dfrac{{5i\left( {2 + 3i} \right)}}{{\left( {2 - 3i} \right)\left( {2 + 3i} \right)}}\) \( = \dfrac{{10i + 15{i^2}}}{{4 + 9}} = - \dfrac{{15}}{{13}} + \dfrac{{10}}{{13}}i.\)

d) \(\dfrac{{5 - 2i}}{i} = \dfrac{{\left( {5 - 2i} \right)i}}{{{i^2}}} \) \(= - \left( {5i - 2{i^2}} \right) = -2 - 5i.\)