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

May 8, 2016

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 $\left(- 1\right)$ can you extract from the expression. Suppose you have $n$ factors of $\left(- 1\right)$ then this can be written as:
$\textcolor{w h i t e}{\text{XXX")(-1)^nxx("some positive value}}$
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))

$\textcolor{w h i t e}{\text{XXX")=(color(red)(-1)) * (color(blue)(-1)) * (color(orange)(-1)) * (color(brown)(-1)) * (color(green)(-1)) * (color(cyan)(-1))xx ("product of only positive values}}$

$\textcolor{w h i t e}{\text{XXX")=(-1)^6xx("product of positive values}}$

$\textcolor{w h i t e}{\text{XXX")=(+1)xx("product of positive values}}$

$\Rightarrow$ a postive result.