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Order of Operations

When several operations crowd one expression, PEMDAS settles who goes first — and saves you from getting two different answers to the same line.

Why we need an agreed order

Take 2 + 3 × 4. Add first and you get 20; multiply first and you get 14. Mathematics cannot tolerate two answers for one line, so everyone agrees on a single order of operations, remembered as PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). By that rule 2 + 3 × 4 = 2 + 12 = 14.

Working a layered expression

Grouping symbols — parentheses ( ), brackets [ ], and braces { } — override everything; whatever they enclose is computed first. With nested parentheses you work from the innermost outward. The same order governs an algebraic expression you must evaluate later, so building the habit now pays off everywhere.

Evaluate:   3 + 2 * (8 - 3)^2 - 10 / 5

  inside parentheses:   8 - 3 = 5
                        3 + 2 * 5^2 - 10 / 5
  exponent:             5^2 = 25
                        3 + 2 * 25 - 10 / 5
  mult / div (L->R):    2*25 = 50 ;  10/5 = 2
                        3 + 50 - 2
  add / sub (L->R):     53 - 2
  =                     51
One line, one answer: parentheses, then exponents, then ×÷, then +− — each left to right.
  1. Clear the innermost grouping symbols first, working outward.
  2. Apply all exponents.
  3. Do multiplication and division as they appear from left to right.
  4. Do addition and subtraction from left to right — minding signs from the previous guide.