Giải bài 3 trang 138 – SGK môn Giải tích lớp 123
Thực hiện các phép tính sau:
a) \(2i\left( 3+i \right)\left( 2+4i \right)\);
b) \(\dfrac{{{\left( 1+i \right)}^{2}}{{\left( 2i \right)}^{3}}}{-2+i}\);
c) \(3+2i+\left( 6+i \right)\left( 5+i \right)\);
d) \(4-3i+\dfrac{5+4i}{3+6i}\).
\(\begin{aligned} a)\,& 2i\left( 3+i \right)\left( 2+4i \right) \\ =&\left( -2+6i \right)\left( 2+4i \right) \\ =&-4-24+\left( -8+12 \right)i \\ =&-28+4i \\ \end{aligned} \)
\(\begin{aligned}b)\, & \dfrac{{{\left( 1+i \right)}^{2}}{{\left( 2i \right)}^{3}}}{-2+i} \\ =&\dfrac{{{\left( 2i \right)}^{4}}\left( -2-i \right)}{5} \\ =&\dfrac{16\left( -2-i \right)}{5} \\ =&-\dfrac{32}{5}-\dfrac{16}{5}i \\ \end{aligned} \)
\(\begin{aligned}c)\, & 3+2i+\left( 6+i \right)\left( 5+i \right) \\ =&3+2i+30-1+\left( 6+5 \right)i \\ =&32+13i \\ \end{aligned} \)
\(\begin{aligned}d)\, & 4-3i+\dfrac{5+4i}{3+6i} \\ =&4-3i+\dfrac{\left( 5+4i \right)\left( 3-6i \right)}{45} \\ =& 4-3i+\dfrac{15+24+\left( -30+12 \right)i}{45} \\ =&4-3i+\dfrac{39}{45}-\dfrac{18}{45}i \\ =&\dfrac{219}{45}-\dfrac{153}{45}i \\ \end{aligned} \)
Ghi nhớ: Cho hai số phức \(z=a+bi\) và \(z'=c+di\).
Khi đó \(z.z'=(ac-bd)+(ad+bc)i\).
\(\dfrac{z}{z'}=\dfrac{z.\overline{z'}}{{{\left| z' \right|}^{2}}}=\dfrac{\left( a+bi \right)\left( c-di \right)}{{{c}^{2}}+{{d}^{2}}}\).