Question

The root(s) of the equation $x^4=16$ is/are :

• A. $2$
• B. $-2$
• C. $2i$
• D. $-2i$

Answer 1

Answer: (A), (B), (C), (D)

$-2i$

Given $x^4=16$
or $x^4=2^4\left(1\right)$
Hence roots will be
$2\cdot e^{i\left(2k\right)\pi /4},k=0,1,2,3$
or $2e^{i0},2e^{i\pi /2},2e^{i\pi },2e^{i3\pi /2}$
$\therefore x=2,2i,-2,-2i$

Alternate Solution:
$x^4=16$
$\Rightarrow x^4-16=0$
$\Rightarrow (x^2{)}^2-(4{)}^2=0$
$\Rightarrow (x^2+4)(x^2-4)=0$
$\Rightarrow x^2+4=0\text{or}{x}^2-4=0$
$\Rightarrow x^2=-4\text{or}{x}^2=4$
$\Rightarrow x=\pm \sqrt{-4}\text{or}x=\pm \sqrt{4}$
$\Rightarrow x=\pm 2i\text{or}x=\pm 2$
Hence four roots of equation are : $2,-2,2i,-2i$