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Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
1.3
Operations on Real
Numbers and
Order of
Operations
Adding Real Numbers
To add two real numbers
1. with the same sign, add their absolute
values. Use their common sign as the sign
of the answer.
2. with different signs, subtract their absolute
values. Give the answer the same sign as
the number with the larger absolute value.
Example
Add.
a. (‒8) + (‒3) = ‒11
Same sign
b. (‒7) + 1 = ‒6
Different signs
9  2 7
  
c.
10  10  10
Different signs
d.
Same signs
(‒12.6) + (‒1.7) = ‒14.3
Subtracting Real Numbers
Subtracting Two Real Numbers
If a and b are real numbers, then
a – b = a + (– b).
Examples
Subtract.
a. –6 – 5 = –6 + (–5) = –11
b. 7 – (–8) = 7 + 8 = 15
c. 4 – 9 = 4 + (–9) = –5
Example
Subtract.
a. 4 ‒ 7 = ‒3
b. ‒8 ‒ (‒9) = 1
c. (–5) – 6 – (–3) = ‒8
Examples
Subtract.
a. 6.9 ‒ (‒1.8) = 6.9 + 1.8 = 8.7
3  4
3 4 1
b.         
4  5
4 5 20
Multiplying Real Numbers
Multiplying Real Numbers
1. The product of two numbers with the same
sign is a positive number.
2. The product of two numbers with different
signs is a negative number.
Examples
Multiply.
a.
4(–2) = –8
b.
‒7(‒5) = 35
c. 9(‒6.2) = ‒55.8
d.
3 1
3 1
3
  

4 7
47
28
Product Property of 0
a · 0 = 0. Also 0 · a = 0.
Example:
Multiply. –6 · 0
–6 · 0 = 0
Example:
Multiply. 0 · 125
0 · 125 = 0
Quotient of Two Real Numbers
The quotient of two numbers with the same sign is
positive.
The quotient of two numbers with different signs
is negative.
Division by 0 is undefined.
Example
Divide.
a. 20  5
4
b. 36  12
3
56
c.
 70
0.8
Examples
36
a. Find the quotient.
12
36
 36  (12)  3
12
3
1
b. Find the quotient. 
2 6
3 5 3 6 3 6 33 2 9
   


2 6 2 5 25
25
5
Simplifying Real Numbers
If a and b are real numbers, and b  0,
a
a
a


b
b
b
Exponents
Exponents that are natural numbers are
shorthand notation for repeating factors.
34 = 3 · 3 · 3 · 3
3 is the base
4 is the exponent (also called power)
Note by the order of operations that exponents
are calculated before other operations.
Example
Evaluate.
a.
(–2)4 = (–2)(–2)(–2)(–2) = 16
b.
‒72 = ‒(7 ·7) = ‒49
Example
Evaluate each of the following expressions.
34 = 3 · 3 · 3 · 3 = 81
(–5)2 = (– 5)(–5) = 25
–62 = – (6)(6) = –36
(2 · 4)3 = (2 · 4)(2 · 4)(2 · 4)= 8 · 8 · 8 = 512
3 · 42 = 3 · 4 · 4 = 48
Square Roots
Opposite of squaring a number is taking the
square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a
number that, when squared, equals a.
Principal Square Roots
The principal (positive) square root is noted
as
a
The negative square root is noted as
 a
Example
Find the square roots.
a. 49  7
b.
5
25

16
4
c.  4   2
d. 49  not a real number
The Order of Operations
Order of Operations
Simplify expressions using the order that follows. If
grouping symbols such as parentheses are
present, simplify expression within those first,
starting with the innermost set. If fraction bars are
present, simplify the numerator and denominator
separately.
1. Evaluate exponential expressions, roots, or
absolute values in order from left to right.
2. Multiply or divide in order from left to right.
3. Add or subtract in order from left to right.
Example
Use order of operations to evaluate each
expression.
a. 7(9)  2(6)  63  12  75
86
14
14
8  3(2)
8  (6)

 


b.
96 3
3
 9  2(3)  9  (6)
Example
Simplify each expression.
a. ‒9 – 5 + 11 – (‒7) = ‒9 + (–5) + 11 + 7 = 4
2
b. 32  8  
5


9


3
 8   5  9
 

 32  8   4
 9   8   4
1 4
5
Example
693
Evaluate:
32
693
32




6  9 3
(9)
6  (3)
9
9
9
1
Write 32 as 9.
Divide 9 by 3.
Add 3 to 6.
Divide 9 by 9.
Evaluating Expressions
Example: Evaluate the expression
4 + (42 – 13)4 – 3.
4 + (42 – 13)4 – 3
= 4 + (16 – 13)4 – 3 Evaluate the exponent
inside the parentheses.
= 4 + (3)4 – 3
= 4 + 81 – 3
Work inside the
parentheses.
Evaluate the exponent.
= 85 – 3
Add.
= 82
Subtract.
Example
Evaluate each of the following expressions.
a.) Find 3x2 when x = 5.
3x2 = 3(5)2 = 3(5 · 5) = 3 · 25 = 75
b.) Find –2x2 when x = –1.
–2x2 = –2(–1)2 = –2(–1)(–1) = –2(1) = –2
Example
Find the value of the expression when
x = 4 and y = ‒3.
43
1
7
x y
4   3




17  x
17  4
3
21
17  4
Example
Evaluate each expression for the given value.
(a) 5x – 2 for x = 8
5(8) – 2 = 40 – 2 = 38
(b) 3a2 + 2a + 4 for a = – 4
= 3(– 4)2 + 2(– 4) + 4
= 3(16) + (– 8) + 4 = 44