📘 complex numbers

1. **State the problem:** Evaluate the expression $$i \left( \frac{1+3i}{1-2i} \right)^n$$ where $i$ is the imaginary unit and $n$ is an integer. 2. **Simplify the fraction inside

1. **Problem statement:** Express the complex numbers $2 + 3i$ and $1 - 2i$ in polar form, then use De Moivre's theorem to evaluate their product $(2 + 3i)(1 - 2i)$. Express the re

1. **State the problem:** Find the cube root of the complex number $81i$. 2. **Recall the formula:** The cube roots of a complex number $z = re^{i\theta}$ are given by

1. **State the problem:** Find the cube root of the complex number $z = -4\sqrt{2} + 4\sqrt{2}i$. 2. **Convert to polar form:** A complex number $z = x + yi$ can be written as $z =

1. **State the problem:** We have two complex numbers represented as vectors in the plane: - $z_2$ with magnitude 8 and angle $\theta$ from the negative x-axis (second quadrant).

1. **State the problem:** We want to express $ (1+i)^{30} $ in the form $ x + iy $ using De Moivre's theorem. 2. **Recall De Moivre's theorem:** For a complex number in polar form

1. **Énoncé du problème :** Vérifier que $(2 - 3i)^2 = -5 - 12i$.

1. The problem asks: If a complex number lies in the third quadrant, then in which quadrant does its conjugate lie? 2. Recall that a complex number $z = x + yi$ lies in the third q

1. **Problem Statement:** Convert the complex number $-5 + 5i$ into polar and exponential form. 2. **Formula and Rules:**

1. **Problem:** Find the modulus and amplitude (argument) of the complex number $1 + 3i$. 2. **Formula:**

1. Given the complex numbers $z_1 = 4\sqrt{3} + 4i$, $z_2 = 1 + i\sqrt{3}$, and $Z = \frac{z_1}{z_2}$. **i. Write $Z$ in the form $a + ib$**

1. Given the complex numbers $z_1 = 4\sqrt{3} + 4i$, $z_2 = 1 + i\sqrt{3}$, and $Z = \frac{z_1}{z_2}$. **i. Write $Z$ in the form $a + ib$**

1. **State the problem:** Convert the complex number $z = \left(\frac{\sqrt{2}}{2} + \sqrt{2}i\right)^8$ into its Cartesian form (a + bi). 2. **Rewrite the complex number:** Note t

1. **Énoncé du problème :** Déterminer la valeur réelle de $x$ dans différents cas liés à des nombres complexes $z$.

1. نبدأ بمسألة التعبير عن العدد المركب $$\sqrt{12} - \sqrt{-4}$$ بالصيغة القطبية. 2. نلاحظ أن $$\sqrt{12}$$ هو عدد حقيقي موجب، و $$\sqrt{-4}$$ هو عدد تخيلي ناتج عن الجذر التربيعي ل

1. **Problem:** Find the modulus $|z|$ of the complex number $z = \frac{2 - i}{2 + i}$. 2. **Formula:** The modulus of a complex number $z = \frac{a}{b}$ is $|z| = \frac{|a|}{|b|}$

1. **State the problem:** Simplify the expression $ (5 - 3i) - (-2 + 5i) $ where $ i = \sqrt{-1} $. 2. **Recall the rule:** Subtracting a complex number means subtracting both its

1. **State the problem:** Simplify the complex expression $$\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^{12}$$. 2. **Recall the formula and rules:** To simplify powers of complex

1. **Problem statement:** Given a complex number $z = x + yi$ satisfying $$\left|\frac{z - (4 - 6i)}{z - (5 - 5i)}\right| = 6,$$

1. The problem is to evaluate the expression $\sin 30^\circ + i \cos 30^\circ$ where $i$ is the imaginary unit. 2. Recall the values of sine and cosine for $30^\circ$:

1. **Problem statement:** Express $ (1+i)^8 $ in the form $ a+bi $. 2. **Formula and rules:** Use the polar form of complex numbers and De Moivre's theorem: