Which of the following expressions is a fourth root of unity? a.) -1 b.) i c.) -i d.) all of the above

Question

asked Jun 13, 2019 91.3k views

2 Answers

Flurpleplurple · Answer 1 · 2019-06-18T12:47:32+0000

We have to evaluate the fourth roots of unity.

For each natural number say 'n', there are exactly 'n' nth roots of unity which is expressed in the form as

$z=e^{(i2\Pi k)/(n)}$

where k=0,1,2,.... n-1

Since we have to evaluate the fourth root of unity.

Therefore, we take k=0,1,2,3 and n=4

So, we get
$z=e^{(i2\Pi k )/(4)}$

$z=e^{(i\Pi k )/(2)}$

Now, For k=0, we get our first root as:

$z=e^{(i\Pi * 0 )/(2)}$

z=e^(0) = 1

First root = 1

Now, for k=1, we get

$z=e^{(i\Pi * 1 )/(2)}$

$z=e^{(i\Pi )/(2)}$

$e^(i\Theta )= \cos \Theta +i\sin \Theta$ (Eulers Formula)

So,
$e^{i(\Pi )/(2)}=\cos (\Pi )/(2)+i\sin (\Pi )/(2)$

= 0 +i = i

So, second root = i

Now, for k=2, we get

$z=e^{(i\Pi * 2)/(2)}$

$z=e^(i\Pi )$

$e^(i\Theta )= \cos \Theta +i\sin \Theta$ (Eulers Formula)

So,
$e^(i\Pi )=\cos \Pi +i\sin \Pi$

= -1 +0 = -1

Third root = -1

Now, for k=3, we get

$z=e^{i(3\Pi )/(2)}$

$e^(i\Theta )= \cos \Theta +i\sin \Theta$ (Eulers Formula)

So,
$e^{i(3\Pi )/(2)}=\cos (3\Pi )/(2)+i\sin (3\Pi )/(2)$

= 0 -i = -i

So, fourth root = -i

Hence, all the fourth roots of unity are 1, i, -1 and -i

Therefore, option D is correct as all the given roots in option A, B and C are the fourth roots of unity.