Use of commutative property in calculating $4765 + (-896) + (896 + 477) + (-4765 + 23)$ The pre-calculus question reads --> state the various properties to easily and mentally compute:
$ 4765+(-896)+ (896+477)+(-4765+23)$ 
It is easy for me to see how the regrouping allows for easy mental math, so I would have said the associative property.  I also see how $-896 + 896$ is the additive inverse property. However, the instructor indicated that this is problem also uses the commutative property and I don't see how this would apply here.
Can someone explain how this can be the commutative prop?  Teacher just indicated that it is clear that those 3 properties were used.  
 A: 4765+(−896)+(896+477)+(−4765+23)
= 4765+ (−896)+ 896 + 477 + (−4765) + 23 [associative property -- we can re-group the additions in any combinations as long as we keep to addition of negatives, not subtraction]
=  4765+ ((−896)+ 896) + 477 + (−4765) + 23 [associative property again]
= 4765 + 0 + 477 + (−4765) + 23 [property of opposite or additive inverse of a real number]
= 4765  + 477 + (−4765) + 23 [property of zero as additive identity]
= 4765  + (−4765) + 477  + 23 [commutative property of addition, re-order a + b = b + a]
= (4765  + (−4765)) + 477  + 23 [associative property again]
= 0 + 477  + 23 [property of opposite or additive inverse again]
= 477  + 23 [property of zero as additive identity again]
= 400 + 70 + 7 + 20 + 3 [place value conventions of number system]
= 400 + 70 + 20 + 7 + 3 [commutative property again]
= 400 + 90 + 10 = 400 + 100 = 500 [addition facts and using properties of base ten system to "carry"]
This all may look insanely obvious to you. Spend a little time helping some kids who are having problems with arithmetic in Grades 1 to 3 and you will learn a new respect for the complexities and subtleties of numbers which you blissfully skim over every day, because you were lucky enough to master all of these skills when you were young.
A: Commutative property allows for the moving of terms left and right. We would use the commutative property to move the $-4765$ term, and could even move the $896$ term so that it is even more clear. In essence, we would use the associate property to remove all parenthesis at the beginning, i.e., $$4765 + (-896) + (896+477) + (-4765 + 23) = \\4765 + (-896) + 896+477 + (-4765) + 23.$$ Then use the commutative property to reorder the terms, i.e., $$4765 + (-896) + 896+477 + (-4765) + 23 = \\ 4765  + (-4765) + 896 + (-896) + 477 + 23.$$ Then use the associative property to regroup the terms, i.e., $$4765  + (-4765) + 896 + (-896) + 477 + 23 = \\ (4765 +(-4765)) + (896+(-896))+(477+23).$$ Then use the additive identity to note the first two associated terms are zero and then the last is easily seen to be $500$, i.e., $$(4765 +(-4765)) + (896+(-896))+(477+23) = 0 + 0 + 500 = 500.$$
