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)

The correct options are
A $2$
B $-2$
C $2i$
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:
$x^4=16$
$x^4-16=0$
$(x^2{)}^2-(4{)}^2=0$ $a^2-b^2=(a+b)(a-b)$
$(x^2+4)(x^2-4)=0$
$x^2+4=0$ $x^2-4=0$
$x^2=-4$ $x^2=4$
$x=\pm \sqrt{-4}$ $x=\pm \sqrt{4}$
$x=\pm 2i$, $x=\pm 2$
Hence four roots of equation are : $2,-2,2i,-2i$