Question

What is the square root of `-64` ?

Answer 1

The square roots of `-64` are `8i` and `-8i` where `i` is the imaginary unit, satisfying `i^2 = -1`.

Explanation:

The square of any Real number is non-negative, so to find a square root for `-64` we need to look beyond the Real number line to Complex numbers.

The imaginary unit `i` is a number whose square is `-1`. So it is a square root of `-1`.

Note that `-i` is also a square root of `-1`, satisfying `(-i)^2 = i^2 = -1`

For any negative number `n` we find:

`(isqrt(-n))^2 = i^2(-n) = (-1)(-n) = n`

So `isqrt(-n)` is a square root of `n`. By convention, this is the principal square root and therefore the one we mean by `sqrt(n)`.

In our particular example:

`sqrt(-64) = isqrt(64) = isqrt(8^2) = 8i`