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

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

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

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

First root = 1

Now, for k=1, we get

(Eulers Formula)

So,

So, second root = i

Now, for k=2, we get

(Eulers Formula)

So,

Third root = -1

Now, for k=3, we get

(Eulers Formula)

So,

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: -8.2562234e+13

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.