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

Question

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

Roman Alvarez · Accepted Answer

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.

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.

Noah Archer · Answer

undefined To find the sign of the expression (8(-2)(14)(-3)(-38))/((-19)(-28)(-27)) without calculating the numerical value, count the number of negative factors in the numerator and denominator. Then, determine if the total count is even or odd. If the count is even, the expression is positive; if it's odd, the expression is negative.