Question

How do you multiply `sqrt(-10)*sqrt(-40)`?

Answer 1

-20

Explanation:

When working in the complex numbers, we need to remember that the rule `sqrta sqrtb = sqrt(ab)` only works if `a` and `b` are not both negative.

We can change `sqrt(-10) = isqrt(10)`.

And then use

`sqrt(-10)sqrt (-40) = isqrt10sqrt(-40) = isqrt (-400) = i(20i) = -20`.

Or we can also rewrite `sqrt(-40) = isqrt40`, so we have:

`sqrt(-10) sqrt(-40) = isqrt(10) i sqrt(40) = i^2 sqrt 400 = -20`.

Answer 2

`sqrt(-10)*sqrt(-40) = +- 20`

Explanation:

`sqrt(-10) * sqrt(-40) = (+-i sqrt(10)) * (+-i sqrt(40))`

`= +-i^2 sqrt(10)*sqrt(40) = +-sqrt(10*40) = +-sqrt(400) = +-20`

The problem here is that `sqrt(-10)` and `sqrt(-40)` are not uniquely defined since `sqrt(-1)` is not uniquely defined.

If `a in RR` and `a > 0` then `sqrt(a)` denotes the positive square root of `a`. It has another square root, viz `-sqrt(a)`.

if `a < 0` then `a` has two pure imaginary square roots which you could call `+-i sqrt(-a)`.

From the perspective of `RR`, the number `i` which we call the square root of `-1` is indistiguishable from `-i`. We cannot pick one of `i` or `-i` as `sqrt(-1)` by saying we want the positive one.