Why do imaginary numbers work this way?

What I know is that (i) is the sqrt of -1. What I dont understand is why (i) squared is -1. If you multiply radicals with similar bases, dont you simply multiply its contents? so wouldnt it be left with the sqrt of 1, and then the answer is 1 not -1??





Im a little confused, please help clear this up.Why do imaginary numbers work this way?
Hi,


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(鈭?)虏 = 鈭? 鈭? = 鈭? = 2





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(鈭?)虏 = 鈭?虏 = 2





Basically squaring and taking the square root are opposites so squaring a square root just cancels out the radical sign.





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(鈭?1 )虏 = 鈭?1 鈭?1 = -1Why do imaginary numbers work this way?
Instead, think of the square root of -1 as -1^(1/2).





Therefore, -1^(1/2) squared is -1^(1/2) * -1^(1/2).





When you multiply exponents with the same base, you add the exponents, which gives you -1^1, -1.
鈭?鈭?=鈭?5=5


鈭歺鈭歺=鈭歺虏=x





in this case x is (-1)


鈭?-1)鈭?-1)=鈭?-1)虏=(-1)





does this help?
No.





When you multiply radicals with same bases you get just the base.





root 3 times root 3 = 3





root 2 times root 2 = 2





root -1 times root -1 = -1