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Thread: Multiplying square roots of negative numbers?

  1. #1
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    Multiplying square roots of negative numbers?

    Which is right (and why)?

    17I+4-2 \cdot \sqrt{-2} \cdot \sqrt{-18} = 17I+4-2I \cdot \sqrt{2} \cdot I\sqrt{18}= \ldots = 16+17I

    or

    17I+4-2 \cdot \sqrt{-2} \cdot \sqrt{-18} = 17*I+4-2 \cdot \sqrt{36}= \ldots = -8+17I

    Actually, according to my calculator, the first one is right, so I really only want to know why.

    Isn't \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}? If so, then \sqrt{-2} \cdot \sqrt{-18} = \sqrt{(-2)(-18)} = \sqrt{36}=6 and the second one is "also" right.

    Note that my calc says:
    \sqrt{(-2)(-18)}=6, and
    \sqrt{-2} \cdot \sqrt{-18}=-6

    Is the conclusion that \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} is only true for a,b \ge 0?

    Thanks
    Last edited by MSUMathStdnt; Mar 10th 2011 at 10:27 PM.
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  2. #2
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    \displaystyle \sqrt{-2}= \sqrt{-1\times 2}= \sqrt{-1}\times\sqrt{ 2}= i\times\sqrt{ 2}

    \displaystyle \sqrt{-18}= \sqrt{-1\times 18}= \sqrt{-1}\times\sqrt{ 18}= i\times\sqrt{ 18}

    \displaystyle \sqrt{-18}\times \sqrt{-2}= i\times\sqrt{ 2}\times i\times\sqrt{ 18}= i^2\times \sqrt{36}= -6
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  3. #3
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    Quote Originally Posted by pickslides View Post
    \displaystyle \sqrt{-2}= \sqrt{-1\times 2}= \sqrt{-1}\times\sqrt{ 2}= i\times\sqrt{ 2}

    \displaystyle \sqrt{-18}= \sqrt{-1\times 18}= \sqrt{-1}\times\sqrt{ 18}= i\times\sqrt{ 18}

    \displaystyle \sqrt{-18}\times \sqrt{-2}= i\times\sqrt{ 2}\times i\times\sqrt{ 18}= i^2\times \sqrt{36}= -6
    OK, I have no problem with that. But why is the rule \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} being violated when a=-2 and b=-18?

    In other words, what's wrong with (the second one of) these:
    \sqrt{-1} \cdot \sqrt{-1} = \imath \cdot \imath=-1
    \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1
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  4. #4
    MHF Contributor Unknown008's Avatar
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    You are right, that rule is not always true.

    Here is a wiki article part about it:

    Square root - Wikipedia, the free encyclopedia
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  5. #5
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    Quote Originally Posted by Unknown008 View Post
    You are right, that rule is not always true.

    Here is a wiki article part about it:

    Square root - Wikipedia, the free encyclopedia
    I have been teaching my Algebra I students that rule for two years now. I can't decide whether I'm upset at the answer or relieved to have the answer.
    Thanks for the link.
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  6. #6
    MHF Contributor Unknown008's Avatar
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    If they haven't learned about complex numbers yet, it's there's still some time
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