One of the values of $(i)^{\large\frac{1}{3}}$ is

Question

(A) $\large\frac{-1+\sqrt 3 i}{2}$

(B) $\large\frac{-1-\sqrt 3 i}{2}$

(C) -i

(D) i

Answer 1

Toolbox:

• $i^3=-i$
• $a^3+b^3=(a+b)(a^2-ab+b^2)$

Let $(i)^{\large\frac{1}{3}}$$=x$

cubing both the sides we get

$i=x^3$

$\Rightarrow\:x^3-i=0$

$\Rightarrow\:x^3+i^3=0$

$\Rightarrow\:(x+i)(x^2-ix-1)$

$\Rightarrow\:x=-i,\:\:or\:\:x=\large\frac{i\pm \sqrt {3}}{2}$