Algebra 1
2.7 Find Square Roots and Compare Real Numbers
Vocabulary:
 Square Root — a number times itself to make the number you started with
 Radicand — the number under the radical symbol
 Perfect Square — the square of an integer

Irrational Number — a number that is not rational
 Real Number — the set of all rational and irrational numbers
EXAMPLE 1: Find Square Roots
Evaluate the expression:
a.
 36 = ±6 The positive and negative square roots of 36 are 6 and - 6
b.
49 = 7
The positive square root of 49 is 7
c.
 4 = -2
The negative square root of 4 is -2
GUIDED PRACTICE:
Evaluate the expression:
a.
 9 = -3
The negative square root of 9 is -3
b.
25 = 5
The positive square root of 25 is 5
c.
 64 = ±8
The positive and negative square roots of 64 are 8 and - 8
d.
 81 = -9
The negative square root of 81 is -9
EXAMPLE 2: Approximate a Square Root
The top of a folding table is a square whose area is 945 square inches. Approximate the side length of the tabletop to the nearest
inch.
SOLUTION:
You need to find the side length s of the tabletop such that s 2 = 945.
This means that s is the positive square root of 945.
You can use a table to determine whether 945 is a perfect square.
Number
Square of
number
28
29
30
31
32
784
841
900
961
1024
As shown in the table, 945 is not a perfect square. The greatest perfect
square less than 945 is 900. The least perfect square greater than 945 is 961.
900 < 945 < 961
900  945  961
30  945  31
Because 945 is closer to 961 than to 900,
945 is closer to 31 to 30.
The side length of the tabletop is about 31 inches.
GUIDED PRACTICE:
1.
Approximate the square root to the nearest integer:
32
You can use a table to determine whether 32 is a perfect square.
As shown in the table, 32 is not a perfect square. The greatest perfect square less than 32 is 25. The least perfect square greater
than 25 is 36.
25 < 32 < 36
25  32  36
5  32  6
Because 32 is closer to 36 than to 25,
2.
32 is closer to 6 to 5.
Approximate the square root to the nearest integer:
103
You can use a table to determine whether 103 is a perfect square.
Number
8
9
10
11
12
Square of
number
64
81
100
121
144
As shown in the table, 103 is not a perfect square. The greatest perfect square less than 103 is 100. The least perfect square
greater than 100 is 121.
100 < 103 < 121
100  103  121
10  103  11
Because 103 is closer to 100 than to 121,
103 is closer to 10 to 11.
3.
Approximate the square root to the nearest integer:
 48
You can use a table to determine whether 48 is a perfect square.
As shown in the table, 48 is not a perfect square. The greatest perfect square less than 48 is 36. The least perfect square greater
than 48 is 36.
-36 < -48 < -49
 36   48   49
6   48  7
Because 48 is closer to 49 than to 36,
4.
48 is closer to -7 than to -6.
Approximate the square root to the nearest integer:
 350
You can use a table to determine whether 350 is a perfect square.
As shown in the table, 350 is not a perfect square. The greatest perfect square less than -350 is -324. The least perfect square
greater than 350 is 361.
-324 < -350 < -361
 324   350   361
18   350  19
Because 350 is closer to 361 than to 324,
 350 is closer to -19 than to -18.
EXAMPLE 3: Classify Numbers
Tell whether each of the following numbers is a real number, a rational number, an irrational number, an integer, or a whole
number:
24, 100, 81
EXAMPLE 4: Graph and order Real Numbers
4
Order the numbers from least to greatest: ,  5, 13, 2.5, 9
3
Solution:
Graph the numbers on a number line.
Read the numbers from left to right: 2.5,  5,
4
, 9, 13
3
GUIDED PRACTICE:
1.
Order the numbers from least to greatest: 
9
,5.2,  20, 7, 4.1, 0
2
Begin by graphing the numbers on a number line.
Read the numbers from left to right: 
9
,  20, 0, 7, 4.1,5.2
2
Classify the following numbers as Real, Rational, Irrational, Integer and/or Whole: 2.5,  5,
2.
-2.5
Real?
Rational?
Irrational?
Integer?
Whole?
yes
yes
no
no
no
yes
no
yes
no
no
yes
yes
no
no
no
yes
yes
no
yes
yes
yes
no
no
5
4/3
9
13
yes
no
4
, 9, 13
3