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$$\sqrt{-5}*\sqrt{-3}=\sqrt{-1*5}*\sqrt{-1*3}$$ $$\sqrt{-1*-1}*\sqrt{5*3}=\sqrt{5*3}$$ $$=\sqrt{15}$$

But we all know that this below is right, $$\sqrt{5}i*\sqrt{3}i=-\sqrt{15}$$

So, please explain the formal result. And any confusion that have been arisen in my mind. This was shown to me by my teacher, and he wanted explanation, and I'm kind of stuck with it. What's true and what's not?

mobifz96
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1 Answers1

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You made the mistake that was made probably hundreds of times on this site:

You assume that $\sqrt{ab}=\sqrt a\sqrt b$ is true for all real (or complex) numbers

In fact, the equality only holds for positive real numbers, and $-5$ is not a positive real number.

5xum
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  • Ok thanks for the info, I arrived at this contradiction independently, and sorry that I didn't knew that previously otherwise I wouldn't have asked this question. – mobifz96 Jun 10 '16 at 15:12
  • Can you point me to any question that explains why a and b have to be positive real. I can't find them. – mobifz96 Jun 10 '16 at 16:45
  • http://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12 – 5xum Jun 10 '16 at 18:01