SOLUTION: to take the 4th root of a -16 isn't possible right? and if it is possible then what would it be?

Question 420170: to take the 4th root of a -16 isn't possible right? and if it is possible then what would it be?
Found 2 solutions by Theo, richard1234:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you can take the root of a negative number if the root is odd.

you cannot take the root of a negative number if the root is even.

if you are taking the fourth root and the number is 16, then the root is 2, because 2^4 = 16

if you are taking the fourth root and the number is -16, then the root is not -2, because (-2)^4 = + 16.

can't be done.

not with real numbers.

they invented imaginary numbers to allow you to do things like take the fourth root of -16, but that's another story.

as long as you are dealing with real numbers, the answer is no, you cannot take the fourth root of -16.

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
The fourth roots of -16 are definitely possible, using complex numbers. Let be a complex number such that z^4 = -16 (we know the magnitude of z must be 2 since 2^4 = 16). We can express z in the form and obtain

(by Euler's formula)

Hence,

Therefore, . From DeMoivre's theorem, . This number must equal -1, so and . This implies --> , where k is any integer. For now, we can say that and the fourth root of -16 is . In a+bi form, this is equal to .
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