ORDER OF OPERATIONS WITH REAL NUMBERS
Simplify the following expression:
Solution:
12 ÷ 3 ∙ 4
16 if you divide first, then multiply
or
1 if you multiply first, then divide
We know that it is not possible to have two answers. That would cause chaos. Therefore, there is a set
order for what operations get simplified before others.
**see below for correct solution**
RULES FOR THE ORDER OF OPERATIONS
1. Simplify any operations WITHIN grouping symbols such as parenthesis ( ), brackets [ ], braces { },
absolute value | |, and above or below fraction bar.
Note: If the expression contains more than one grouping symbol, then work from the innermost
symbols to outermost symbols.
2. Simplify any exponents/powers and radicals.
3. Simplify division and multiplication as they occur from left to right.
4. Simplify subtraction and addition as they occur from left to right.
NOTE: The order of operations still apply within the grouping symbols.
**The correct solution to the above expression, 12 ÷ 3 ∙ 4, is 16. Following the order of operation you
divide first because that is what occurred first in this particular expression. Please be aware that
multiplication will not always be done before division. The order will vary with each problem.
EXPONENTIAL NOTATION
Exponential notation refers to the form 53 in which 5 is the base and 3 is the power/exponent.
54 means 5 ∙ 5 ∙ 5 ∙ 5 = 625
(-5)4 means -5 ∙ -5 ∙ -5 ∙ -5 = 625
-54 means - (5 ∙ 5 ∙ 5 ∙ 5) = -625
-(5)4 means - (5 ∙ 5 ∙ 5 ∙ 5) = -625
-(-5)4 means - (-5 ∙ -5 ∙ -5 ∙ -5) = -625
Note that grouping symbols changes the meaning of what the problem is stating to do.
EXAMPLES:
The final solution is in red.
The bold print is the terms being used in the step.
1) Simplify -3(-4) - 2(5)
12 - 10
2
multiply
add/subtract
2) Evaluate -10 ÷ 2(5)
-10 ÷ 2(5)
divide
-5(5)
multiply
-25
NOTE: In this expression the parenthesis is playing the role of multiplication symbol and not a grouping
symbol. Therefore, you can rewrite the expression as -10 ÷ 2 ∙ 5 and continue the steps.
3) Perform the indicated operations:
24 ÷ 3 ∙ 4 - 25 ÷ 23 ∙ 7
24 ÷ 3 ∙ 4 - 32 ÷ 8 ∙ 7
8 ∙ 4 - 32 ÷ 8 ∙ 7
32 - 32 ÷ 8 ∙ 7
32 - 4 ∙ 7
32 - 28
4
4) Simplify
simplify exponents
multiplication/division
moving from left to right
simplify addition/subtraction
5 + 7(9 - 12)
5 + 7(9 - 12)
5 + 7(-3)
5 + -21
-16
5) Simplify
24 ÷ 3 ∙ 4 - 25 ÷ 23 ∙ 7
simplify within parenthesis
multiply
add/subtract
-(-2)3(-0.5)2(0.1)2
-(-8)(0.25)(0.01)
8(0.25)(0.01)
0.02
simplify exponents
multiply
6) Simplify
5 - 4[3 + 2(6 - 13)]
5 - 4[3 + 2(6 - 13)]
5 - 4[3 + 2(-7)]
5 - 4[3 - 14]
5 - 4[-11]
5 + 44
49
7) Simplify
simplify within parenthesis
simplify multiplication within brackets
simplify add/sub within brackets
multiply
add/sub
8 - 3[9 - 2(-7 - 5)2]
8 - 3[9 - 2(-7 - 5)2]
8 - 3[9 - 2(-12)2]
8 - 3[9 - 2(144)]
8 -3[9 - 288]
8 - 3[-279]
8 + 837
845
8)
Simplify
simplify within parenthesis
simplify exponents
multiply within brackets
add/sub within brackets
multiply
add/sub
48 - 4[32 - (5 - 11)2 + 2]3 ÷ 2 ∙ -3
48 - 4[32 - (5 - 11)2 + 2]3 ÷ 2 ∙ -3
48 - 4[32 - (-6)2 + 2]3 ÷ 2 ∙ -3
48 - 4[32 - 36 + 2]3 ÷ 2 ∙ -3
48 - 4[-2]3 ÷ 2 ∙ -3
48 - 4[-8] ÷ 2 ∙ -3
48 + 32 ÷ 2 ∙ -3
48 + 16 ∙ -3
48 - 48
0
9) Simplify.
simplify within parenthesis
simplify exponents within brackets
add/sub within brackets
simplify exponents
multiply
divide
multiply
add/sub
(8 - 5)3 - |52 - 43| ÷ (1 - 4)
(8 - 5)3 - |52 - 43| ÷ (1 - 4)
(3)3 - |25 - 64| ÷ (-3)
(3)3 - |-39| ÷ (-3)
(3)3 - 39 ÷ (-3)
27 - 39 ÷ (-3)
27 + 13
40
simplify within parenthesis and absolute value
simplify within absolute value
simplify the absolute value
simplify power/exponent
divide
add/sub
10) Perform the indicated operations
-12 ∙ 5 ∙ 1 ∙ 4
3 8 1
when dividing fractions, convert to multiplication
and take the reciprocal of second fraction
-240 = -10
24
11) Evaluate.
-12 ÷ 3 ∙ 1 ÷ 1
5 8
4
multiply across numerator and denominator
and reduce fraction when possible
3(7 - 5)3 ÷ 6 + (3 - 8)
5 - |-4| - 33 ÷ 9 ∙ 3
simplify the numerator and denominator following the order of operations.
simplifying the numerator
simplifying the denominator
3(7 - 5)3 ÷ 6 + (3 - 8)
3(2)3 ÷ 6 + (-5)
within parenthesis
3(8) ÷ 6 + (-5)
powers/exponents
24 ÷ 6 + (-5)
multiplication
4 + (-5)
division
-1
add/subtract
5 - |-4| - 33 ÷ 9 ∙ 3
5 - 4 - 33 ÷ 9 ∙ 3 the absolute value
5 - 4 - 27 ÷ 9 ∙ 3 powers/exponents
5-4-3∙3
division
5 - 4 - 27
multiplication
-26
add/subtract
recall that once you have simplified the numerator and denominator, you still need to finish the problem
by reducing the fraction if possible.
final solution is -1/-26 or 1/26 or 0.038