Question

Which of the following expressions is a fourth root of unity?

a.) -1
b.) i
c.) -i
d.) all of the above

Answer 1

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

$z=e^{\frac{i2\Pi k}{n}}$

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 $z=e^{\frac{i2\Pi k }{4}}$

$z=e^{\frac{i\Pi k }{2}}$

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

$z=e^{\frac{i\Pi \times 0 }{2}}$

$z=e^{0} = 1$

First root = 1

Now, for k=1, we get

$z=e^{\frac{i\Pi \times 1 }{2}}$

$z=e^{\frac{i\Pi }{2}}$

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

So, $e^{i\frac{\Pi }{2}}=\cos \frac{\Pi }{2}+i\sin \frac{\Pi }{2}$

$= 0 +i = i$

So, second root = i

Now, for k=2, we get

$z=e^{\frac{i\Pi \times 2}{2}}$

$z=e^{i\Pi }$

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

So, $e^{i\Pi }=\cos \Pi +i\sin \Pi$

$= -1 +0 = -1$

Third root = -1

Now, for k=3, we get

$z=e^{i\frac{3\Pi }{2}}$

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

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

$= 0 -i = -i$

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.

Answer 2

Answer:

The CORRECT answer is D. All of the above

Step-by-step explanation:

I just took the test and got it correct!

(see screen shot for proof)