Question

Why is `sqrt(-2) sqrt(-3) != sqrt((-2) * (-3))` ?

Answer 1

The "rule" `sqrt(ab) = sqrt(a)sqrt(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) = -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.