Forme exponentielle - Corrigé

Énoncé

Soit \(z_1=4\text e^{-\frac{2i\pi}{5}}\) et \(z_2=\frac{1}{3}\text e^{\frac{4i\pi}{9}}\) . Déterminer une forme exponentielle des nombres complexes suivants.

1. \(z_1 z_2\)

2. \(\dfrac{z_1}{z_2}\)

3. \(z_1^4\)

4. \(\dfrac{z_2^2}{z_1^5}\)

Solution

1. On a :  \(z_1 z_2= 4\text e^{\frac{-2i\pi}{5}} \times \frac{1}{3}\text e^{\frac{4i\pi}{9}}= \frac{4}{3}\text e^{i\left(\frac{-2\pi}{5}+\frac{4\pi}{9}\right)}= \dfrac{4}{3}\text e^{\frac{2i\pi}{45}}\) .

2. On a :   \(\dfrac{z_1}{z_2}= \dfrac{4\text e^{\frac{-2i\pi}{5}} }{ \frac{1}{3}\text e^{\frac{4i\pi}{9}} }= 12 \text e^{i\left(\frac{-2\pi}{5}-\frac{4\pi}{9}\right)}= 12 \text e^{-\frac{38i\pi}{45}}\) .

3. On a : \(z_1^4 = \left( 4\text e^{\frac{-2i\pi}{5}} \right)^4= 4^4\text e^{\frac{-8i\pi}{5}}= 256 \text e^{\frac{-8i\pi}{5}}= 256\text e^{\frac{2i\pi}{5}}\) .

4. On a :   \(\dfrac{z_2^2}{z_1^5}= \dfrac{ \left( \frac{1}{3}\text e^{\frac{4i\pi}{9}} \right)^2 }{ \left( 4\text e^{\frac{-2i\pi}{5}} \right)^5 }= \dfrac{1}{3^2 \times 4^5} \dfrac{ \text e^{\frac{8i\pi}{9}} } { \text e^{\frac{-10i\pi}{5}} }= \dfrac{1}{3^2 \times 4^5} \dfrac{ \text e^{\frac{8i\pi}{9}}}{ 1 }= \dfrac{1}{9\,216} \text e^{i \frac{8\pi}{9} }\) .