Download Notes 1 - Henrico

Order of Operations rules for arithmetic and algebra
that describe what sequence to
follow to evaluate an expression
involving more than one operation
The Rules
Step 1: Do operations inside grouping symbols such
as parentheses (), brackets [], and braces {},
and operations separated by fraction bars.
Parentheses within parentheses are called
nested parentheses (( )).
Step 2: Evaluate Powers (exponents) or roots.
Step 3: Perform multiplication or division in order
by reading the problem from left to right.
Step 4: Perform addition or subtraction in order by
reading the problem from left to right.
Order of Operations - WHY?
Imagine if two different people wanted to evaluate the
same expression two different ways...
#1 does each step left to right:
21  6  3 5
21  6  3  5
27  3 5
27  3 5
9 5
 45
#2 uses the order of operations
The rules for order of
operations exist so
that everyone can
perform the same
consistent operations
and achieve the same
results. Method 2 is
the correct method.
21  6  3 5
21  6  3  5
21  2  5
21  2  5
21 10
 31
Order of Operations - WHY?
• Can you imagine what it would be like if
calculations were performed differently by
various financial institutions?
• What if doctors prescribed different doses
of medicine using the same formulas but
achieving different results?
Order of Operations: Example 1
Evaluate without grouping symbols
54  6 18 2
54  6  18  2
Divide.
9 18  2
This expression has no parentheses and no
exponents.
• First solve any multiplication or
division parts left to right.
• Then solve any addition or subtraction
parts left to right.
Multiply.
9  36
 45
Add.
The order of operations must be followed
each time you rewrite the expression.
Order of Operations: Example 2
Expressions with powers
25  6
• Firs,t solve exponents
(powers).
25  6
• Second, solve multiplication
or division parts left to right.
2
2
Exponents (powers)
2  25  6
Multiply.
• Then, solve any addition or
subtraction parts left to right.
50  6
Subtract.
 44
The order of operations must be followed
each time you rewrite the expression.
Order of Operations: Example 3
Evaluate with grouping symbols
3  4  8  2
• First, solve parts inside grouping
symbols according to the order
of operations.
2
3  42  8  2
3  42  6
3 16  6
48  6
8
Grouping
symbols
• Solve any exponent (Powers).
Subtract.
Exponents (powers)
Multiply.
Divide.
• Then, solve multiplication or
division parts left to right.
• Then solve any addition or
subtraction parts left to right.
The order of operations must be followed
each time you rewrite the expression.
Order of Operations: Example 4
Expressions with fraction bars
Work above the
fraction bar.
3 4
2  (18  4)
2
Work below the
fraction bar.
Exponents (powers)
3 4 2
2  (18  4)
Grouping symbols
Multiply.
316
2  (14)
Add.
48
16
Simplify
: Divide.
4816  3
Order of Operations: Example 5
Evaluate variable expressions
( x  y  5)  n
Evaluate when x=2, y=3, and n=4:
1) Substitute in the values for the variables
3
Inside
grouping
symbols:
6
Exponents (powers)
(2  33  5)  42  6
Add.
(2  27  5)  42  6
Subtract.
Continue
with the
rest:
2
(29  5)  42  6
Exponents (powers)
24  4 2  6
Subtract.
24 16  6
Add.
86
 14
The order of
operations
must be
followed each
time you
rewrite the
expression.