When simplifying numerical expressions, you must follow rules called the order of
operations. They work as follows:
1) Perform operations within grouping symbols ― parentheses ( ), brackets [ ],
braces { }, radicals , absolute value bars
, or in the numerator or
denominator of a fraction.
2) Simplify all exponents (including square roots, cube roots, or other radicals).
3) Multiply and/or divide terms in order from left to right.
4) Add and/or subtract terms in order from left to right.
An east way of remembering the order of operations for simplifying numerical
expressions is with the abbreviation PEMDAS:
Parentheses
example 1
Compute:
The first expression to simplify is the one in parentheses:
The rules for order of operations applies to this expression, where any multiplication
must happen first (since there are no exponents):
Now these numbers can be added:
The remainder of the original expression can be simplified now:
The answer is −8.
Absolute Value
example 2
What is the value of the following expression?
The absolute value of a number is its distance from zero, which can be seen visually
on a number line:
|x|
–x
|x|
0
x
What that means mathematically is, the positive and negative forms of any number
have the same absolute value:
From the above expression,
rewritten as:
Notice that only the signs of each number changed, but the subtraction of the third
term is still there. Only minus signs inside absolute value symbols are changed.
The expression can then now be simplified:
The answer is 7.
Exponents
example 3
Compute:
An exponent indicates how many of something is being multiplied together. So −4
raised to the third power is the same as:
The only operation being applied here is multiplication, so you can multiply from left
to right (and remember that a negative times a negative equals a positive):
The answer is −64.
(Note: When a negative number is raised to a odd power, or in other words, when the
exponent is an odd number, the result will be negative. But when a negative number is
raised to an even power, the result will be positive.)
example 4
What is the value of the expression 3(2 – 4)2 + 3?
A. −33
B. −9
C. 15
D. 39
In the problem, just go through the order of operations (PEMDAS).
The answer is C.
Evaluating Algebraic Expressions
example 5
If x = 2 and y = 4, what is the value of the following expression?
This problem requires you to substitute numbers into each variable and then simplify
the expression. Without any numbers to substitute, this expression is as simplified as
it can be, but when we put 2 in for x, and 4 in for y, the numerical expression we get
can be simplified, using the rules for the order of operations of course:
The answer is A.
Name ______________________________
Simplify each expression:
1) (6 + 2)2
2) 33 – 32 × 3
3) 8 ÷ 2 × 3 ÷ 6
4) 8 − 5 + 5 − 8
5) 8 · 22 + 5 – (23 – 4)
6) 15 ÷ 3 + 4(3 – 2)2
7) (2 + (23 ÷ 22))2
8) 30 ÷ (4 + 2(7 – 4)2 – 3 · 4)
2
9) 4 · (6 – 2) + (3 + 1)
10)
42 − 62
(8 − 2) ÷ 2
– 23
Simplify each expression:
11) 3 + 4 · 5 – 6 · 7 ÷ 3
13)
3 + (2 × 11)
(11 + 4) ÷ 3
12)
(5 − 2) 2 + (8 − 6) 2
14) 2 + [3 + (27 ÷ 9)]2
15) 52 ÷ 5 + 22 · 6 – 20 ÷ 5
16)
23 − 32
+ (6 – 4) ÷ 2
4−5
Simplify each expression if a = 1, b = 2, c = 3, x = 4, y = 5, and z = 6:
18) (a – x)2 ÷ (b + 1)
17) b − z + a − y
19)
(b − z)
( x − a) 2
+
c
x
2
20)
( y + 2) − ( y − 2)