In this section, we will learn how to operate with proper order of operations when addition (subtraction), multiplication (devision), exponents and parentheses are present in the expression.
The Order of Operations:
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When evaluating an expression involving addition, subtraction, multiplication and division which has no parentheses or exponents, we first perform, from left to right, all of the multiplications and divisions. Then, from left to right, the additions and subtractions. If there are parts of the expression set off by parentheses, what is within the parentheses must be evaluated first.
PE(MD)(AS) is an easy way remember the order of operations. This means that the order is: Parentheses, Exponents, Multiplication and Division (taken together from left to right), and finally, Addition and Subtraction (taken together from left to right).
Examples:
a) $3 – 2(-4 + 7)$
$ = 3 – 2(3) $
$= 3 – 6 $
$= -3$
b) $-3 – 2(-2\cdot 6 – 5)$
$= -3 – 2(-12 – 5)$
$= -3 – 2(-17)$
$= -3 – (-34) $
$= -3 + 34 $
$= 31 $
c) $-3(-2\cdot 7 – (-5)(4)\div 2)$
$= -3(-14 – (-20)\div 2) $
$= -3(-14 – (-10))$
$= -3(-14 + 10) $
$= -3(-4) $
$= 12$
d) $6 \div 2 \times 3 $
$= 3 \times 3 $
$= 9$. Note: $6 \div 2 \times 3 \neq 6 \div 6 = 1$
e) $-2(3-1)^2 – (8-2^2)\div 4$
$= -2(2)^2 – (8 – 4) \div 4$
$= -2(4) – 4 \div 4$
$= -8 – 1$
$= -9$
In relation to the order of operations PE(MD)(AS), the absolute value symbol is treated as a parenthesis, and so, what is inside has the priority over other operations.
Example:
a) $10\cdot\dfrac{|3^2-3|}{2}+3$
$= 10\cdot \dfrac{|9-3|}{2}+3$
$= 10\cdot \dfrac{6}{2}+3 $
$= 10\cdot 3+3$
$= 33$
Practice Problems:
(a) $3\cdot 2^3 – 6 \cdot 5$
(b) $(-5\cdot 2 + 3)^2 – 1$
(c) $2 \cdot (8- 3^2) $
(d) $(3^3 + 5) \div 4 – 4(7 – 2)$
(e) $2\cdot |-4| + 5^2 – 3$
Answer Key: (a) $-6$ (b) $48$ (c) $-2$ (d) $-12$ (e) $30$
For more detailed explanation, please read: Arithmetic|Algebra Chapter 1.
