Why is sqrt(-2) sqrt(-3) != sqrt((-2) * (-3))√−2√−3≠√(−2)⋅(−3) ?

1 Answer
George C.
Jan 9, 2017

The "rule" sqrt(ab) = sqrt(a)sqrt(b)√ab=√a√b does not hold all of the time - especially when it comes to negative or complex numbers.

Explanation:

Here's another example:

1 = sqrt(1) = sqrt((-1)*(-1)) != sqrt(-1)*sqrt(-1) = -11=√1=√(−1)⋅(−1)≠√−1⋅√−1=−1

This kind of thing happens because every non-zero number has two square roots and the one we mean when we write sqrt(...) can be a slightly arbitrary choice.

So long as you stick to a, b >= 0 then the rule holds:

sqrt(ab) = sqrt(a)sqrt(b)

When you get to deal with complex numbers more fully (in precalculus?) then it may become clearer.

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