Question

where are the properties faulty?

Answer 1

\(\large\color{black}{ \displaystyle i=i^{1}=i^{4\times(1/4)}=\left(i^4\right)^{1/4}=\sqrt[4]{1}=1 }\)

Answer 2

It's like \(-1=\sqrt{(-1)^2}=\sqrt 1= 1\),
whenever we square an expression, we need to get rid of extraneous roots by back substitution.
There are 4 fourth-roots of 1, \(\pm 1~ and~ \pm i\).

Answer 3

Oh, in the example you proposed, i think
\(\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}\)
doesn't work if √a and √b are both complex. that is what i heard

Answer 4

but yeah it is a similar accomplishment

Answer 5

but even if i²=±1, then i still get i=1

Answer 6

i think i got what you are trying to say

Answer 7

the even root of a constant is presumed to be positive, but that doesn't apply to i, so you get ±1.

But, how would then (i^4)^(1/4)= 1 not be true?

Answer 8

\[\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}\] is true if \(a,b>0\) otherwise it is not

Answer 9

There are 4 fourth-roots of 1, ±1 and ±i. +1 is one of them. The choice depends on the context.

Answer 10

Alright, yes, thanks:)

Answer 11

You're welcome! :)