SOLUTION: what is the square root of -6 multiplied by the square root of -6

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Question 1108124: what is the square root of -6 multiplied by the square root of -6
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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The other tutor's answer is not exactly correct. 

In complex domain,  sqrt%28-6%29+%2A+sqrt%28-6%29 = sqrt%2836%29  has two (TWO) values: +6  and  -6.


    Why  ?  - can you ask me.

              Isn't it the product of any two complex numbers is defined by an UNIQUE way ?



    Yes, it is true, the product of two complex numbers is defined by an UNIQUE way.  There is no doubt.


    But  in complex domain,  sqrt%28-6%29 has itself  TWO values:  6i  and  -6i.

    They are TWO absolutely symmetric complex instances with equal rights to exist and with equal rights to represent  sqrt%28-6%29.


    If you multiply  sqrt%286%29%2Ai  by  sqrt%286%29%2Ai,  you  will get  -6.

    If you multiply  -sqrt%286%29%2Ai  by  -sqrt%286%29%2Ai,  you  will get  -6, too.

    But if you multiply  sqrt%286%29%2Ai  by  -sqrt%286%29%2Ai,  you  will get  6. 

    Symmetrically, if you multiply  -sqrt%286%29%2Ai  by  sqrt%286%29%2Ai,  you  will get  6, too. 


Therefore, there is NO CONTRADICTION in getting two values, +6  and  -6,  of  the product  sqrt%28-6%29%2Asqrt%28-6%29.


If you want to avoid getting of two values when you multiply  sqrt%28-6%29  by  sqrt%28-6%29,  you MUST to point EXACTLY, 

which of the two values of sqrt%28-6%29  you mean  as one of the two possible instances for each multiplier.