Simplifications - Corrigé

Énoncé

Simplifier les écritures suivantes.

• \(z_1=\dfrac{8}{\text e^{-\frac{3i\pi}{7}}}\)
  
• \(z_2=5 \times \text e^{-\frac{i\pi}{4}} \times \text e^{\frac{5i\pi}{9}}\)
• \(z_3=2 \times \dfrac{\text e^{-\frac{2i\pi}{3}}}{\text e^{-\frac{3i\pi}{11}}}\)
• \(z_4=\left(\dfrac{\text e^{-\frac{2i\pi}{5}}}{\text e^{\frac{i\pi}{2}}}\right)^6\)
• \(z_5=\dfrac{\left(\text e^{\frac{5i\pi}{9}}\right)^7}{\left(3\text e^{-\frac{i\pi}{5}}\right)^3}\)

Solution

• \(z_1=z_1=\dfrac{8}{\text e^{-\frac{3i\pi}{7}}} = 8 \times \text e^{\frac{3i\pi}{7}}\)
  
• \(z_2=5 \times \text e^{-\frac{i\pi}{4}} \times \text e^{\frac{5i\pi}{9}} = 5\text e^{i\left(-\frac{\pi}{4}+\frac{5\pi}{9}\right)}= 5\text e^{-\frac{11i\pi}{36}}\)
  
• \(z_3=2 \times \dfrac{\text e^{-\frac{2i\pi}{3}}}{\text e^{-\frac{3i\pi}{11}}}= 2\text e^{i\left(-\frac{2\pi}{3}+\frac{3\pi}{11}\right)}= 2\text e^{-\frac{13i\pi}{33}}\) .
  
• \(z_4=\left(\dfrac{\text e^{-\frac{2i\pi}{5}}}{\text e^{\frac{i\pi}{2}}}\right)^6 = \dfrac{\text e^{-\frac{12i\pi}{5}}}{\text e^{\frac{6i\pi}{2}}}= \dfrac{\text e^{-\frac{12i\pi}{5}}}{\text e^{3i\pi}}= \text e^{i\left(-\frac{12\pi}{5}-3\pi\right)}= \text e^{\frac{-27i\pi}{5}}\)
  
• \(z_5= \dfrac{\left(\text e^{\frac{5i\pi}{9}}\right)^7}{\left(3\text e^{-\frac{i\pi}{5}}\right)^3}= \dfrac{\text e^{\frac{35i\pi}{9}}}{3^3\text e^{-\frac{3i\pi}{5}}}= \frac{1}{27} \dfrac{\text e^{\frac{-i\pi}{9}}}{\text e^{-\frac{3i\pi}{5}}}= \frac{1}{27} \text e^{i\left(-\frac{\pi}{9}+\frac{3\pi}{5}\right)}= \frac{1}{27} \text e^{\frac{22i\pi}{45}}\)