Rešite 1+i/1-sqrt{3i}

$$\frac { 1 + i } { 1 - \sqrt { 3 } i }$$

Ovrednoti

$$\sqrt{3}\left(-\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}+\frac{1}{4}i\right)\approx -0,183012702+0,683012702i$$

Koraki rešitve

Racionalizirajte imenovalec \frac{1+i}{1-i\sqrt{3}} tako, da pomnožite števec in imenovalec s 1+i\sqrt{3}.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{\left(1-i\sqrt{3}\right)\left(1+i\sqrt{3}\right)}$$

Razmislite o \left(1-i\sqrt{3}\right)\left(1+i\sqrt{3}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1^{2}-\left(-i\sqrt{3}\right)^{2}}$$

Izračunajte potenco 1 števila 2, da dobite 1.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1-\left(-i\sqrt{3}\right)^{2}}$$

Razčlenite \left(-i\sqrt{3}\right)^{2}.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1-\left(-i\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Izračunajte potenco -i števila 2, da dobite -1.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1-\left(-\left(\sqrt{3}\right)^{2}\right)}$$

Kvadrat vrednosti \sqrt{3} je 3.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1-\left(-3\right)}$$

Pomnožite -1 in -3, da dobite 3.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{1+3}$$

Seštejte 1 in 3, da dobite 4.

$$\frac{\left(1+i\right)\left(1+i\sqrt{3}\right)}{4}$$

Delite \left(1+i\right)\left(1+i\sqrt{3}\right) s/z 4, da dobite \left(\frac{1}{4}+\frac{1}{4}i\right)\left(1+i\sqrt{3}\right).

$$\left(\frac{1}{4}+\frac{1}{4}i\right)\left(1+i\sqrt{3}\right)$$

Uporabite distributivnost, da pomnožite \frac{1}{4}+\frac{1}{4}i s/z 1+i\sqrt{3}.

$$\frac{1}{4}+\frac{1}{4}i+\left(-\frac{1}{4}+\frac{1}{4}i\right)\sqrt{3}$$

Realni del

$$\frac{1 - \sqrt{3}}{4} = -0,1830127018922193$$