How do you find three cube roots of −1?

1 Answer
Shwetank Mauria
Feb 22, 2017

Three cube roots of 1 are {−1,12−√32i,12+√32i}

Explanation:

Let x be the cube root of −1, then we have x⋅3=−1

or x3+1=0

⇔x3+x2−x2−x+x+1=0

or x2(x+1)−x(x+1)+1(x+1)=0

or (x+1)(x2−x+1)=0

Hence either x+1=0 i.e. x=−1, or x2−x+1=0.

So one root is x=−1 and for other two roots of x2−x+1=0, we proceed as follows:

x2−x+1=0⇔x2−2×x×(12)+(12)2−(12)2+1=0

or (x−12)2+34=0

i.e. (x−12)2−(√32i)2=0

i.e. (x−12+√32i)(x−12−√32i)=0

i.e. x=12−√32i or x=12+√32i

Hence, three cube roots of 1 are {−1,12−√32i,12+√32i}

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