Question

Question #586a5

Answer 1

Your question doesn’t seem logical.

Explanation:

Neither `3sqrt(-64)` or `sqrt(-64)` can be simplified to real numbers.
The square root of a negative number is always an imaginary number. The examples can be simplified to have the lowest value radicand, but this will still be an imaginary number.

From examples:

`3sqrt(-64) =>` `3sqrt(64 *( -1) )` `=>` `24sqrt(-1)`

`sqrt(-64) =>` `sqrt(64*(-1)` `=> 8sqrt(-1)`

These can also be expressed as `24i` and `8i` or `(0 + 24i)` and `(0 + 8i)`, where `i` is the imaginary unit `i =` `sqrt(-1)`.
These are known as pure imaginary numbers.

Answer 2

Because a negative number raised to an odd power gives a negative result.

Explanation:

`(-3)^3 = -27`, so `root(3)(-27) = -3`

`(-2)^5 = -32`, so `root(5)(-32) = -2`

But for even powers, both positives and negatives raised to an even power give a positive result.

`(5)^2 = 25` and also `(-5)^2 = 25`

And any other number raised to the power `2` will give a positive answer (except `0^2 = 0`).

So no number can be raised to the power `2` and give a negative result.

(Until we introduce a new kind of numbers that lets us do that.)