766.

Algebarski oblik

TEKST ZADATKA

Naći realni i imaginarni deo kompleksnog broja:

(−1+i32i)2\bigg(\frac{-1+i\sqrt3}{2i}\bigg)^2
REŠENJE ZADATKA

Razdvojiti razlomak:

(−12i+i32i)2(−12i+32)2\bigg(\frac{-1}{2i}+\frac{i\sqrt3}{2i}\bigg)^2 \\ \bigg(-\frac{1}{2i}+\frac{\sqrt3}{2}\bigg)^2

Primeniti formulu za kvadrat zbira:(a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2

(−12i)2+2⋅(−12i)⋅32+(32)214i2−32i+34\bigg(-\frac1{2i}\bigg)^2+2\cdot\bigg(-\frac1{2i}\bigg)\cdot\frac{\sqrt3}2+\bigg(\frac{\sqrt3}2\bigg)^2 \\ \frac1{4i^2}-\frac{\sqrt3}{2i}+\frac34

Racionalisati izraz.

14i2−32i⋅ii+3414i2−3i2i2+34\frac1{4i^2}-\frac{\sqrt3}{2i}\cdot\frac{i}{i}+\frac34 \\ \frac1{4i^2}-\frac{\sqrt3i}{2i^2}+\frac34

Pošto je i2=−1i^2=-1 dobija se:

−14+3i2+3412+32i-\frac1{4}+\frac{\sqrt3i}{2}+\frac34 \\ \frac12+\frac{\sqrt3}{2}i

Realni deo kompleksnog broja 12+32i\frac12+\frac{\sqrt3}{2}ije12,\frac12, a imaginarni 32.\frac{\sqrt3}{2} .

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