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Order of Operations

Arithmetic Order of Operations (BEDMAS)

Addition:

a + b

Subtraction:

ab

Multiplication:

a × b or a*b

Division:

a ÷ b or a/b

Exponent:

a b or a^b

Note that      − a = 0 − a = ( − 1)× a,      and     ab = a + ( − b )


Order of Operations

One need only think of a toddler with two operations to do, eat lunch and wash face, to realise that the order in which the operations are done makes a tremendous difference to the result.

When evaluating numerical or algebraic expressions, we need to know the order in which addition, subtraction, multiplication, division and exponents are carried out. For all numerical or algebraic expressions, the order of evaluation is ( BEDMAS ):

B rackets and Parentheses First Priority
E xponents Second Priority
D ivision Third Priority
M ultiplication Third Priority
A ddition Fourth Priority
S ubtraction Fourth Priority

If an expression involves two or more operations at the same level of priority, those operations are done from left to right.


Example 1A.

Click on the question marks to see the following examples done step-by-step. Keep the following order of operations in mind when studying these examples:

Brackets, Exponents, Division and Multiplication, Addition and Subtraction

1.) 2 * 4 - 3 + 7 = 8-3+7=5+7=12

2.) 2 * 4 - (3 + 7) = 8-10=-2

3.) 3 * 6 - 5 * 2 =18-10=8

4.) 3 (7+2*5)=3*(7+10)=3*17=51

5.) (5-2)(1+9)=3*10=30

6.) 7-2(8-6)=7-2*2=7-4=3

7.) 8+15/3=8+5=13

8.) (10+2^3 *3)+4^2=(10+8*3)+16=(10+24)+16=34+16=50

 

More Examples

 

 

Example 1B.

When brackets occur within brackets, solve the expression inside the inner-most brackets first.

[4 + 2(3 + 2 × 4)] ÷ 2 = [4 + 2(3 + 8)] ÷ 2 = [4 + 2(11)] ÷ 2 = [4 + 22] ÷ 2 = 26 ÷ 2 = 13

 

Example 1C.

When a quotient is written as a ratio the numerator and the denominator are evaluated first. That is, we treat the numerator and denominator as if they were inside brackets. For example,

(a+b)/(c+d)

 

Example 1D.

To calculate -a2, first find the square of a, a2. Then take its negative. So -a2 is the negative of a2.

To calculate (-a)2, square -a.

For example, -32 = -9. But (-3)2 = 9.

(While the above is the correct order of operations, note that Excel evaluates -a2 as (-a)2.)

Exercise 1.

Now try some of these exercises:

Compute:

=

 

If an expression involves functions such as

, ex, ln(x), sin(x) or cos(x) they have to be evaluated first, before they can be raised to a power, multiplied, divided, added or subtracted. They are given First Priority.


Example 2.

2+square root (15-2*3) = 2 + square root (15-6) = 2+ square root 9 = 2 + 3 = 5

More Examples

 

Exercise 2.

Now try some of these exercises. (The notation sqrt(a) is used for square root of aand exp(x) for ex.)

Compute, rounding any decimal answers to 2 d.p.:

=

Worksheet

Order of Operations Index | Evaluating Algebraic Expressions >>

 

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