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The problem is to find $\sqrt{(-1)\cdot(-1)}$
Approach 1 - $\sqrt{(-1)\cdot (-1)} = \sqrt{(-1)^2} = -1$
Approach 2 - $\sqrt{(-1)\cdot (-1)} = \sqrt{1} = 1$

Which is correct and why?

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    This already has an answer on the site. – 2017-02-05
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    I can't see it, mind linking it? – 2017-02-05
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    Related: https://math.stackexchange.com/q/461695/115115, and from there http://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12, http://math.stackexchange.com/questions/49169/why-sqrt-1-times-1-neq-sqrt-12 – 2017-02-05
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    Approach 1 is the correct answer. Hint: 1 x 1 is not a square. – 2017-02-05

2 Answers 2

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When dealing with the imaginary unit, you need to be careful using properties such as:

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

This holds for all $ a,b \in \mathbb{R}_+ \cup \{0\} $, but not for negative real numbers. Therefore, the wrong is the first:

$$ \sqrt{ (-1)\cdot (-1)} \neq i \cdot i $$

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    I can understand this because we could get a negative answer in this way for every square root, but what is the fundamental mistake? Saying you can't do that is not really helpful – 2017-02-05
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$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

where a ≥ 0, b ≥ 0 Or a ≥ 0, b < 0

But NOT a < 0, b < 0

So applying in on a = b = -1 is invalid.