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

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

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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    $\begingroup$ you cant split the roots when numbers inside are negative $\endgroup$
    – asddf
    Commented Feb 3, 2017 at 0:33
  • $\begingroup$ Yes that is what Arnold said you both are correct. $\endgroup$ Commented Feb 3, 2017 at 0:36

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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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  • $\begingroup$ I see. So the middle part of the string of inequalities is false then? $\endgroup$ Commented Feb 3, 2017 at 0:36
  • $\begingroup$ Yes, it is wrong! $\endgroup$
    – Arnaldo
    Commented Feb 3, 2017 at 0:37
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    $\begingroup$ 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. $\endgroup$ Commented Feb 3, 2017 at 0:38
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    $\begingroup$ @Oliver821: feel free to ask everything you want. $\endgroup$
    – Arnaldo
    Commented Feb 3, 2017 at 0:39
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    $\begingroup$ @Arnoldo in the future yes but for now everything is clear. $\endgroup$ Commented Feb 3, 2017 at 0:41

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