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So I encountered an exercise in my algebra textbook and it is somewhat paradoxical. Here is the exercise:

$1 = \sqrt{1}$ = $\sqrt {(-1)(-1)}$ = $\sqrt{(-1)}$ $\sqrt{(-1)}$ = $i*i = i^2 = -1$

I think it has to do with the first step of the problem. The number $\sqrt{1}$ shouldn't simplify to just 1. Is it possible that $\sqrt{1}$ can also be $± 1$?

Edit: I wasn't aware that someone else had already asked a similar question to what I just asked.

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    you cant split the roots when numbers inside are negative – 2017-02-03
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    Yes that is what Arnold said you both are correct. – 2017-02-03

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$$\sqrt{ab}=\sqrt{a}\sqrt{b}$$

That is true only if $a\ge 0$ and $b\ge 0$.

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    I see. So the middle part of the string of inequalities is false then? – 2017-02-03
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    Yes, it is wrong! – 2017-02-03
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    Okay now it makes sense! I knew it had to do with separating the square roots but I wasn't so sure so I decided to ask it as a question. – 2017-02-03
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    @Oliver821: feel free to ask everything you want. – 2017-02-03
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    @Arnoldo in the future yes but for now everything is clear. – 2017-02-03