Order of Operations with Real Numbers

ORDER OF OPERATIONS WITH REAL NUMBERS

Simplify the following expression: 12 ÷ 3 ∙ 4

Solution: 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 multiply
2 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 - 25÷ 23∙ 7 simplify exponents
24÷ 3 ∙ 4 - 32 ÷ 8 ∙ 7
8 ∙ 4 - 32 ÷ 8 ∙ 7 multiplication/division
32 - 32 ÷ 8 ∙ 7 moving from left to right
32 - 4 ∙ 7
32 - 28 simplify addition/subtraction
4

4) Simplify 5 + 7(9 - 12)
5 + 7(9 - 12) simplify within parenthesis
5 + 7(-3) multiply
5 + -21 add/subtract
-16

5) Simplify -(-2)3(-0.5)2(0.1)2

-(-8)(0.25)(0.01) simplify exponents
8(0.25)(0.01) multiply

0.02

6) Simplify 5 - 4[3 + 2(6 - 13)]
5 - 4[3 + 2(6 - 13)]simplify within parenthesis
5 - 4[3 + 2(-7)] simplify multiplication within brackets
5 - 4[3 - 14] simplify add/sub within brackets
5 - 4[-11]multiply
5 + 44add/sub
49

7) Simplify8 - 3[9 - 2(-7 - 5)2]

8 - 3[9 - 2(-7 - 5)2] simplify within parenthesis
8 - 3[9 - 2(-12)2]simplify exponents
8 - 3[9 - 2(144)] multiply within brackets
8 -3[9 - 288] add/sub within brackets
8 - 3[-279] multiply
8 + 837add/sub
845

8) Simplify 48 - 4[32 - (5 - 11)2 + 2]3 ÷ 2 ∙ -3

48 - 4[32 - (5 - 11)2 + 2]3 ÷ 2 ∙ -3 simplify within parenthesis
48 - 4[32 - (-6)2+ 2]3 ÷ 2 ∙ -3 simplify exponents within brackets
48 - 4[32 - 36 + 2]3 ÷ 2 ∙ -3add/sub within brackets
48 - 4[-2]3 ÷ 2 ∙ -3simplify exponents
48 - 4[-8] ÷ 2 ∙ -3multiply
48 + 32 ÷ 2 ∙ -3 divide
48 + 16 ∙ -3 multiply
48 - 48 add/sub
0

9) Simplify. (8 - 5)3 - |52 - 43| ÷ (1 - 4)

(8 - 5)3 - |52 - 43| ÷ (1 - 4) simplify within parenthesis and absolute value
(3)3 - |25 - 64| ÷ (-3)simplify within absolute value
(3)3 - |-39| ÷ (-3)simplify the absolute value
(3)3- 39 ÷ (-3) simplify power/exponent
27 - 39 ÷ (-3) divide
27 + 13 add/sub
40

10) Perform the indicated operations -12 ÷3 ∙1 ÷1
5 8 4
-12 ∙5 ∙1 ∙4 when dividing fractions, convert to multiplication
3 8 1 and take the reciprocal of second fraction

-240 =-10 multiply across numerator and denominator
24 and reduce fraction when possible

11) Evaluate. 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) 5 - |-4| - 33 ÷ 9 ∙ 3
3(2)3 ÷ 6 + (-5) within parenthesis 5 - 4 - 33 ÷ 9 ∙ 3 the absolute value
3(8) ÷ 6 + (-5) powers/exponents 5 - 4 - 27 ÷ 9 ∙ 3 powers/exponents
24 ÷ 6 + (-5) multiplication 5 - 4 - 3 ∙ 3 division
4 + (-5) division 5 - 4 - 27 multiplication
-1 add/subtract -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