Question

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

Answer 1

The answer is all of the above.

Answer 2

• all of the above

Explanation:

If in doubt, multiply them out.

(-i)^4 = i^4 = 1 . . . . . results for ±i

(-1)^4 = 1^4 = 1 . . . . . results for ±1

An even power of a negative number is equivalent to the same power of the opposite (positive) number. The even number of minus signs cancel in the final product.

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In trig form, unity is 1∠(n·360°) . . . . . for n = any integer

Then the 4th root is ...

1^(1/4) ∠(n·360°/4) = 1 ∠(n·90°) . . . . for n = 0...3 (angles repeat after that)

That is, the roots are ...

• 1∠0° = 1
• 1∠90° = i
• 1∠180° = -1
• 1∠270° = -i

Of these, all but the first are listed among your answer choices. The appropriate selection is all of the above. (recognizing that not all of the appropriate choices are "above" that answer selection).