[tex]\text i = \sqrt{-1} \\\\ \text i^1 = \text i \\\\ \text i^2=-1\\\\ \text i^3=-\text i\\\\ \text i^4 = 1 \to (\text{A partir da{\'i} a sequ{\^e}ncia repete})[/tex]
4) Expresse z na forma a + b.i :
item a)
[tex]\displaystyle \text z = \frac{3-4\text i}{2\text i} \\\\ \text{sabendo que i}=\sqrt{-1}, \text{vamos racionalizar, assim :} \\\\ \text z = \frac{(3-4\text i)}{2.\text i}.\frac{\text i}{\text i} \\\\\\ \text z = \frac{3\text i-4\text i^2 }{2\text i^2} \\\\\\ \text z = \frac{3\text i-4.(-1)}{2.(-1)} \\\\\\ \text z = \frac{4+3\text i}{-2} \\\\ \underline{\text{Portanto}}: \\\\\\ \huge\boxed{\text z =-2- \frac{3\text i}{2} \ }\checkmark[/tex]
item b)
[tex]\displaystyle \text z = \frac{1}{\text i}+\frac{1}{1+\text i} \\\\\\ \text i = \sqrt{-1} \\\\\ \underline{\text{Raconalizando}}: \\\\ \text z = \frac{1}{\text i}.\frac{\text i}{\text i} + \frac{1}{(1+\text i)}.\frac{(1-\text i)}{(1-\text i)}\\\\\\ \text z = \frac{\text i}{\text i^2}+\frac{(1-\text i)}{1^2-\text i^2} \\\\\\ \text z = \frac{\text i}{-1}+\frac{(1-\text i)}{1-(-1)} \\\\\\ \text z = -\text i+\frac{1-\text i}{2} \\\\\\ \text z = \frac{-2\text i+1-\text i}{2}[/tex]
[tex]\huge\boxed{\ \text z=\frac{1}{2}-\frac{3\text i}{2}\ }\checkmark[/tex]
5)
item a)
[tex]\text i^{99} \\\\ \text i^{96}.\text i^{3} \\\\ (\ \text i^4 \ )^{24} .(-\text i) \\\\ (1)^{44}.(-\text i) \\\\ \huge\boxed{\text i^{99}= \ -\text i \ }\checkmark[/tex]
item b)
[tex](1-\sqrt{3}.\text i)^{3}\\\\ \undelrine{\text{usando o seguinte produto not{\'a}vel}}: \\\\ (\text a-\text b)^3=\text a^3-3\text{a}^2.\text b+3\text{a.b}^2-\text b^3 \\\\ \text{no caso}: \\\ \text a = 1 \ , \ \text b = \sqrt{3}.\text i \\\\\ \text{substituindo}: \\\\ (1-\sqrt{3}.\text i)^3=1^3-3.1^2.\sqrt{3}.\text i +3.1.(\sqrt{3}.\text i)^2-(\sqrt{3}.\text i)^3 \\\\ (1-\sqrt{3}\text i)^3=1-3\sqrt{3}.\text i+3.1.3.\text i^2-3\sqrt{3}.\text i^3[/tex]
[tex](1-\sqrt{3}\text i)^3=1-3\sqrt{3}.\text i+3.1.3.(-1)-3\sqrt{3}.(-\text i ) \\\\ (1-\sqrt{3}.\text i)^3=1-3\sqrt{3}.\text i-9+3\sqrt{3}.\text i\\\\\ (1-\sqrt{3}\text i)^3 = 1-9 \\\\ \huge\boxed{(1-\sqrt{3}\text i)^3=-8\ }\checkmark[/tex]
