What is the effect of computing xy on the size of existing relative errors in the stored values of x and y?

This is what I have done:

We have x̃=(1+ε_x)x and ỹ=(1+ε_y)y so that

x̃ỹ =(1 + ε_x + ε_y + ε_x*ε_y) xy

=> (x̃ỹ/xy) - 1 = ε_x + ε_y + ε_x*ε_y

=> ε_xy = ε_x + ε_y + ε_x*ε_y

Do I now get rid of ε_x*ε_y because that error is too small to include? (Is this the correct reason or is it something else?)

So that I'm left with

ε_xy = ε_x + ε_y? If this is correct, what does it mean?