Another question...
Which of the following expressions is a sixth root of unity?
1. i
2. -1
3. -i
4. All of the above are correct
I appreciate the help.
e^0 i = cos 0 + i sin 0 = 1 that does not help much
e^2 pi i = cos 2pi + i sin 2pi = 1
so
1^(1/6) e^(2 pi/6)i
2 pi/6 = pi/3 = 60 deg
so cos theta + i sin theta
0 deg 1
60 deg
120 deg
180 deg -1
240 deg
300 deg
none of these is along the i axis so i and -i are out
only 2. which is -1 at 180 degrees is in
thank you so much!!!
You are welcome :)
To determine which expression is a sixth root of unity, we need to understand what a sixth root of unity is.
A sixth root of unity is a complex number that, when raised to the power of 6, equals 1. In other words, it is a solution to the equation x^6 = 1.
Let's evaluate each of the given expressions to see which one satisfies this condition:
1. i
When we raise i to the power of 6, we get (i^2)^3, which is equal to (-1)^3, which equals -1. Since -1 is not equal to 1, i is not a sixth root of unity.
2. -1
When we raise -1 to the power of 6, we get (-1)^6, which is equal to 1. Therefore, -1 is a sixth root of unity.
3. -i
When we raise -i to the power of 6, we get (-i)^6, which is equal to (i^2)^3, which is equal to (-1)^3, which equals -1. Since -1 is not equal to 1, -i is not a sixth root of unity.
Based on our evaluations above, option 2, -1, is the only expression that satisfies the condition of being a sixth root of unity. Therefore, the correct answer is option 2.
I hope this explanation clarifies the concept of sixth roots of unity and how to determine which expression satisfies the condition.