Usage of "cancel" and "cancel out"

Question

It seems that cancel and cancel out can be used in the same way. For example, A cancels B or A cancels out B. In which context is one preferred over the other?

Answer 1

Cancel out is a Phrasal Verb, extending the sense of the verb cancel with the completive aspectual sense of out.

Some other examples:

• stress out 'lose effective control of oneself under stress, and because of it'
• wink out 'be unable to stay awake any longer, no matter how interesting the conversation is, nor how pressing the social obligation to stay awake may be'
• burn out
  • [literal; of fires] 'cease burning, usu. by deprivation of fuel'
  • [metaphoric; of humans] l'ose effective motivation, usu. by deprivation of reward'

The out produces the complete cancellation sense, but that's also available with cancel alone since it can mean 'to delete'. This is especially true with a Reciprocal construction like each other.

All of the following, for example, are grammatical, and equivalent in meaning.

• Each cancelled the other.
• Each cancelled the other out.
• Each cancelled out the other.
• They cancelled each other.
• They cancelled each other out.
• They cancelled out each other.

As can be seen, Phrasal Verbs have interesting and complex syntax, especially the transitive ones.

Answer 2

When two things cancel each other out it means they "neutralize" each other, resulting in "zero." An example of this would be a sound cancelled out by another sound with inverted phase; by combining such two sounds, they cancel each other out and no sound is audible - the result is zero.

On the other hand, when something cancels something else, one of the two elements of cancellation still remains "active" even after the process. A user can cancel a computer process, for example. The user doesn't "disappear" after it happens.

TL;DR:

If A cancels B = A remains existent, B becomes nonexistent.
If A cancels out B = A and B become nonexistent.