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Date: ___________
Review of Real Numbers & Radicals
RATIONAL
The Real Number Properties
If a, b, and c are real numbers then the following properties hold:
INTEGER
(1)
a  b  b  a and a b  b  a
Commutative
IRRATIONAL
WHOL
E
(2)
a c
b  ch
c
a  bh
 c and a  c
b  ch
c
a  bh
c
Associative
(3)
THE REAL NUMBERS
a e
b  cj  a b  a  c
Distributive
1. Which of the following is not an integer?
(1)  9
(2) 0
(3)
2. Which of the following is a rational number?

(1) 7
(2)
2
1
15
(3) 
(4)
3
2
6
3
(4)
3. Which of the following numbers is an integer but not a whole number?
12
(1) -4
(2)
(3) 
3
20
(4) 9
4. Which of the following numbers is a rational number but not an integer?
3
15
(1) 12
(2) 
(3)
7
3
(4) 6
5. Determine the value of each variable shown below to the nearest hundredth. Then, plot the points on
the number line.
125
 90
3
b
c
d
a  12
2
63
4
-5
-4
-3
-2
-1
0
1
2
3
4
5
6. Give an example of an irrational number and explain why it is irrational.
7. Which property is illustrated by the equation 3x – 6y = 3(x - 2y)?
(1) Associative
(2) Distributive
(3) Commutative
(4) Inverse
8. Which of the following properties is illustrated by the equation
3  (5  4)  (3  5)  4 ?
(1) Commutative
(2) Associative
(3) Distributive
(4)Identity
9. Identify each of the following statements with the correct property being illustrated from the following
list. Write the letter that corresponds to the property.
A.
B.
C.
D.
Associative Property
Identity Property
Commutative Property
Distributive Property
(a) (18  3)  5  18  (3  5)
______
(f) 4  (r  s)  (r  s)  4
______
(b) 2  0  2
______
(g) (m  0)  p  m  p
______
(c) 2(b  4)  2b  8
______
(h) 10( x  3)  10 x  30
______
(d) 1  a  a
______
(i) p 1  p
______
Division Property of Square Roots
a
b

a
where
a  0 and b  0
b
35
Recall:
Radical: symbol used to represent
the square root
 
Coefficient: integer in front of the
radical symbol
Addition and Subtraction of Square Roots
 3
Radicand: number underneath the
radical symbol
 5
If a and b are two positive real numbers, then in general:
a  b  a b
The same can be said about subtraction of square roots.
We can combine those without like radicands by simplifying radicals first. Once we have like radicands, we can
combine the radicals using addition or subtraction.
10. Expressed in simplest radical form, 99 is equivalent to
(1) 11 2
(2) 4 11
(3) 3 11
(4) 9 11
____
11. Express each of the following square root expressions in simplest radical form.
(a) 12 27
(b) 8 24
(c) 8 8
(d) 6 300
(e) 14 12
(f) 3 98
12. Express each of the following sums and differences in simplest radical form.
(a) 18 6  4 6  6
(b)
13. Which of the following is equivalent to
(1) 2 3
(2) 8 3
48  27
(c) 3 20  2 45
12  48 ?
(3)  36
(4) 2 3
14. In the accompanying diagram of right triangle ABC, find the length of AC in simplest radical
form.
A
2
B
?
C
5
15. Find the following products, express in simplest radical form.
 
(a) 3 5  4 10



(b)  4 3  10 21

 
(c) 2 6 5 12

Unit 1 Review
Homework
Matching
a.
{…-3, -2, -1, 0, 1, 2, 3,...}
1. Whole numbers
1. ______
b.
7+6=6+7
2. Commutative property
2. ______
c.
b + (-b) = 0
3. Associative property
3._______
d
{0, 1, 2, 3, ...}
4. Distributive property
4._______
e.
x 1  x
5. Integers
5. ______
f.
6  (5  9)  (6  5)  9
6. Inverse property
6._______
g.
2(7  8)  2(7)  2(8)
7. Identity property
7._______
8.
Which of the following is an irrational number?
(1) 16
(2) 6
(3) 1.01011
(4) -7.3333
9. Which property is used to justify the following factorization: 5x2  15x  5x  x  3 ?
(1) Distributive
(2) Identity of Multiplication
(3) Associative
(4) Commutative of Multiplication
10. In the calculation 8  4  52  10 which of the following operations is performed first?
(1) Dividing by 10
(3) Squaring 5
(2) Adding 4
(4) Multiplying by 4
11. Which of the following equations illustrates the identity property for addition?
(1) 1 x  x
(3) 5x  5x  0
(2) 7 x  3 y  3 y  7 x
12. The expression
(1) 5 3
(4) 3   2 x   3  2  x
27  12 is equivalent to
(2) 13 3
(3) 5 6
(4) 39
13. The expression 5 8  3 2 is equivalent to
(1) 7
(2) 7 2
(3) 2 6
(4) 34
14. In a right triangle, the length of the hypotenuse is 12 and the length of one leg is 8.
What is the length of the other leg?
(1) 208
(2) 80
(3) 20
(4) 4
15.
Which property is illustrated by the equation 3x – 6y = 3(x - 2y)?
(1) associative
(2) commutative
(3) distributive
(4) closure
16. The graphs of four different relations are shown below. Which graph represents a relation
that is not a function?
y
y
y
y
(1)
(2)
(3)
(4)
x
x
x
x
17. Tell whether each number is rational or irrational.
a) 0.36
b) 8
c)
18.
Find the value of (3a ) 2 , if a = -3.
19.
If t = 3 and s = 0, find the value of
121
1 2
t s.
3
d) -5.28
e) 
20.
Find the value of
ab 2 c 3 , if a  4 b  3 c  2 .
21. Write each expression in simplest radical form.
a.
27
b.
108
c.
48
d.
44
e.
125
f.
600