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sciencesolve eNotes educator| Certified Educator

The terms `sqrt(-5)` and `sqrt(-15),` separately, are invalid. Using the properties of multiplication of square roots, yields:

`sqrt(a)*sqrt(b) = sqrt(a*b)`

Reasoning by analogy, yields:

`sqrt(-5)*sqrt(-15) = sqrt((-5)*(-15))`

Performing the multiplication of two negative numbers yields:

`sqrt(-5)*sqrt(-15) = sqrt 75 => sqrt(-5)*sqrt(-15) = 5sqrt 3`

Hence, performing the multiplications `sqrt(-5)*sqrt(-15) = 5sqrt 3.`

rcmath eNotes educator| Certified Educator

`sqrt(-5)*sqrt(-15)=isqrt(5)*isqrt(15)=i^2*sqrt(5*15)=i^2*5sqrt(3)=-1*5sqrt(3)=`

`-5sqrt(3)`

sqrt(-1) only exist in the complex number system. It is defined as the imaginary number i and i^2=-1 not 1.

Wilson2014 eNotes educator| Certified Educator

One thing that was overlooked is that the `sqrt(5*15)` can be plus or minus so the full, complete answer is `+- 5*sqrt (3)` .    

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