GMAT Maths Arithmetic - Order of Operation

The standard order of operations, or precedence, is expressed in the following chart.

exponents and roots
multiplication and division
addition and subtraction

In the absence of parentheses, horizontal fraction lines, a bar over a radicand, or other symbols of grouping, do all exponents and roots first. Stacked exponents must be done from the top down. After taking all exponents and roots, then do all multiplication and division. Finally, do all addition and subtraction.

It is helpful to treat division as multiplication by the reciprocal and subtraction as addition of the opposite. Thus 3/4 = 3 ÷ 4 = 3 • ¼ and 3 − 4 = 3 + (−4), that is, the sum of positive three and negative four.

If an expression involves parentheses, then do the arithmetic inside the innermost pair of parentheses first and work outward. Root symbols have a bar (called vinculum) over the radicand which acts as a symbol of grouping: \sqrt{1+3}+5=\sqrt4+5=2+5=7. A horizontal fractional line also acts as a symbol of grouping: \frac{1+2}{3+4}+5=\frac37+5.

Example:

1. Evaluate subexpressions contained within parentheses, starting with the innermost expressions. (Brackets [ ] are used here to indicate what is evaluated next.)
(4+10/2)/9=(4+[10/2])/9=[4+5]/9=[9/9]=1 \,
2. Evaluate exponential powers; for iterated powers, start from the right:
2^{3^2}=2^{[3^2]}=[2^9]=512 \,
3. Evaluate multiplications, divisions and of, starting from the left:
1/2\,\text{ of }\,8/2\times3=[8/4]\times3=[2\times3]=6 \,
4. Evaluate additions and subtractions, starting from the left:
7-2-4+1=[7-2]-4+1=[5-4]+1=[1+1]=2 \,
5. Evaluate negation on the same level as subtraction, starting from the left:
-3^2=-[3^2]=-9 \,

The operations in a multistep expression must be completed in a specific order. This particular order can be
remembered as PEMDAS. In any expression, evaluate in this order:

  • P Parentheses/grouping symbols first
  • E then Exponents
  • MD Multiplication/Division in order from right to left
  • AS Addition/Subtraction in order from left to right