Question

How can we find the sign, positive or negative, of `(8(-2)(14)(-3)(-38))/((-19)(-28)(-27))`, without determining numerical answer?

Answer 1

An even number of multiplications and/or divisions by negative values implies a positive result.
An odd number of multiplications and/or divisions by negative values implies a negative result.

Explanation:

One way to think of this is to ask how many factors of `(-1)` can you extract from the expression. Suppose you have `n` factors of `(-1)` then this can be written as:
`color(white)("XXX")(-1)^nxx("some positive value")`
which will be positive if `n` is even
and negative if `n` is odd.

For the given example:
`color(white)("XXX")(8(color(red)(-2))(14)(color(blue)(-3))(color(orange)(-38)))/((color(brown)(-19))(color(green)(-28))(color(cyan)(-27))`

`color(white)("XXX")=(color(red)(-1)) * (color(blue)(-1)) * (color(orange)(-1)) * (color(brown)(-1)) * (color(green)(-1)) * (color(cyan)(-1))xx ("product of only positive values")`

`color(white)("XXX")=(-1)^6xx("product of positive values")`

`color(white)("XXX")=(+1)xx("product of positive values")`

`rArr` a postive result.