Order of Operations and Properties

Objective: Use exponents, multiplication/division, and addition/subtraction correctly.

Grouping symbols set priority; then evaluate exponents, multiplication and division from left to right, and addition and subtraction from left to right. Multiplication does not always precede division; they share a level. Likewise addition and subtraction share a level. Thus 18÷3×2=12 and 10−6+2=6. The expression −3² means −(3²)=−9, whereas (−3)²=9. A fraction bar groups its entire numerator and denominator. Explicit parentheses can remove ambiguity in a lesson or calculator input.

Objective: Identify when operations can be rearranged.

The commutative property changes order: a+b=b+a and ab=ba. The associative property changes grouping of addition or multiplication: (a+b)+c=a+(b+c). The distributive property connects multiplication with addition: a(b+c)=ab+ac, including negative factors. These properties do not license swapping the operands of subtraction or division. For instance 3(2x−4)=6x−12. Identifying the exact property helps justify each algebraic rewrite rather than relying on a pattern that may fail for some operations.

Objective: Combine terms that share variable factors.

Equivalent expressions have the same value for every input in their common domain. Like terms contain exactly matching variable parts and exponents, so 3x+5x=8x but 3x+5x² cannot combine into 8x³. Distributing and combining like terms can reveal equivalence: 2(x+4)+3x=5x+8. A proposed rewrite can be disproven by one counterexample input, but checking a few inputs alone does not prove two expressions equal for all inputs. Domain restrictions must also be preserved: canceling a factor in (x²−1)/(x−1) yields x+1 only for x≠1.