Why is the Square Root of Negative One equal to i?

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Yes, the "principle square root" of negative 1 is i, however in search of the answer x²=-1, the quadratic nature of this problem will pose 2 possible correct answers, hence both i and -i are correct.

Similarly, a cubic function x³=1 should have 3 possible roots. I suggest you try to figure out this answer using your newfound knowledge though this video before the spoiler alert down below.

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  / sciencenme  







SPOILER ALERT!!!

x³=1 has 3 possible roots.
You can think of the answer as "What 3 same angles, if added together will equal a full circle?".

So If you break a circle down into 3 equal parts, then:
1∠0° is a possible answer (1∠0° x 1∠0° x 1∠0° = 1∠0° = 1)
1∠120° is a possible answer (1∠120° x 1∠120° x 1∠120° = 1∠360° = 1) (1∠120° = - 0.5 + (√3)/2 i )
1∠240° is a possible answer (1∠240° x 1∠240° x 1∠240° = 1∠720° = 1) (1∠240° = - 0.5 - (√3)/2 i )

And yes I know "x" is cross multiply. There is only so much that can be done though typing before the solution becomes too convoluted to interpret.