Question

What is the square root of -50 times the square root of -10?

Answer 1

`sqrt(-50)*sqrt(-10) = -10sqrt(5)`

Explanation:

This is slightly tricky, since `sqrt(a)sqrt(b) = sqrt(ab)` is only generally true for `a, b >= 0`.

If you thought it held for negative numbers too then you would have spurious 'proofs' like:

`1 = sqrt(1) = sqrt(-1*-1) = sqrt(-1)sqrt(-1) = -1`

Instead, use the definition of the principal square root of a negative number:

`sqrt(-n) = i sqrt(n)` for `n >= 0`, where `i` is 'the' square root of `-1`.

I feel slightly uncomfortable even as I write that: There are two square roots of `-1`. If you call one of them `i` then the other is `-i`. They are not distinguishable as positive or negative. When we introduce Complex numbers, we basically pick one and call it `i`.

Anyway - back to our problem:

`sqrt(-50) * sqrt(-10) = i sqrt(50) * i sqrt(10) = i^2 * sqrt(50)sqrt(10)`

`= -1 * sqrt(50 * 10) = -sqrt(10^2 * 5) = -sqrt(10^2)sqrt(5)`

`= -10sqrt(5)`