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Unit #1
Real Numbers and Their
Properties
Properties of Order
For all real numbers a, b, and c
Trichotomy Property
Either a < b, a > b, or a = b
Transitive Property
If a < b and b < c, then a < c
Addition Property
If a < b, then a + c < b + c
Multiplication Property
If a < b, and if, c > 0, then ac < bc
If a < b, and if, c < 0, then ac > bc
Absolute Value
a if a  0
For all real numbers a, a  
a
if
a<0

Write an equivalent expression without
using absolute value bars
3 2
1. 3  
2.
4. x  6
5. 4  2 3
2
3. x  7 ;x  7
6. 3    7
Properties of Absolute
Value
For
1. aallreal
0 numbers a and
2. ba  a
3. a b  ab
5. a  b  a  b
Triangle Inequality
Property
a a
4.
 ;b  0
b b
Properties of Exponents
Let m and n be rational numbers.
The results are valid for all positive
numbers a and b.
Rational Exponents
For all int egers m, all posit ive int egers n,
1/n
and all real numbers a for which a is a
real number : m/n
1/n m
a  a 
Perform each operation mentally
3
4.4
1.
3
1.1
2.  .25
2
 44 
2
3. If x  20, what is x ?
2
6
Perform the indicated operations. Write the
answer using only positive exponents.
Assume that all variables represent positive
real numbers and that variables used as
exponents represent rational numbers.
1.  x y
3/2

1/5 10
 2x  1
3
 2x  1
5
3.
2.
y 
z 2 4
 4xy z 
4.  2 3 
 x yz 
5 2
1/2
Perform the indicated operations. Write the
answer using only positive exponents.
Assume that all variables represent positive
real numbers and that variables used as
exponents represent rational numbers.
2 2
5.
4
2
1/ 4
x

5/ 4
6.
3 y 2
7.
y
2x
x 
3 y 2
y
2x
; y0
 x14  x 
8. 
2
7
  2x 

5 9
; y  3




1/7
Perform the indicated operations. Write the
answer using only positive exponents.
Assume that all variables represent positive
real numbers and that variables used as
exponents represent rational numbers.
 3a   ab
 25a b 
3 2
9.
1
14 4 1/2
2 3
 1 4   1 5  
10.  x y    xy  
4
2



 

2