Simplification d'écritures - Corrigé

Énoncé

Simplifier les écritures suivantes.

\(z_1=3\text e^{-\frac{i\pi}{3}} \times 8\text e^{\frac{i\pi}{4}}\)

\(z_2=\dfrac{-2\text e^{i\frac{\pi}{2}}}{\text e^{\frac{i\pi}{7}}}\)

\(z_3=\left(2\text e^{\frac{i\pi}{4}}\right)^6\)

\(z_4=\dfrac{\left(\text e^{{i\pi}}\right)^5}{\left(4\text e^{\frac{i\pi}{9}}\right)^2}\)

Solution

On a : \(\displaystyle \begin{align*}z_1& =3\text e^{-\frac{i\pi}{3}} \times 8\text e^{\frac{i\pi}{4}}= 24\text e^{-\frac{i\pi}{3}+\frac{i\pi}{4}}= 24\text e^{i\left(-\frac{\pi}{3}+\frac{\pi}{4}\right)}= 24\text e^{-\frac{i\pi}{12}}.\end{align*}\)

On a : \(\begin{align*}z_2& = \dfrac{-2\text e^{i\frac{\pi}{2}}}{\text e^{\frac{i\pi}{7}}}=-2\text e^{i\frac{\pi}{2}-\frac{i\pi}{7}}=-2\text e^{i\left(\frac{\pi}{2}-\frac{\pi}{7}\right)}=-2\text e^{\frac{5i\pi}{14}}.\end{align*}\)

On a : \(\begin{align*}z_3& = \left(2\text e^{\frac{i\pi}{4}}\right)^6= 2^6\text e^{\frac{6i\pi}{4}}= 64\text e^{\frac{3i\pi}{2}}.\end{align*}\)

On a : \(\begin{align*}z_4& = \dfrac{\left(\text e^{{i\pi}}\right)^5}{\left(4\text e^{\frac{i\pi}{9}}\right)^2} = \frac{\text e^{ 5i\pi}}{4^2\text e^{\frac{2i\pi}{9}}}= \frac{1}{16}\text e^{5i\pi-\frac{2i\pi}{9}}= \frac{1}{16}\text e^{i\left(5\pi-\frac{2\pi}{9}\right)}= \frac{1}{16}\text e^{\frac{43i\pi}{9}}= \frac{1}{16}\text e^{\frac{7i\pi}{9}}.\end{align*}\)