1. Lesson 2.7 Finding Square Roots and Compare Real Numbers
Objective: You will find square roots and compare real numbers.
Why? So you can find side lengths of geometric shapes.

2. Evaluate the expression.
EXAMPLE 1
Find square roots
Evaluate the expression. a.  – + 36 = 6
The positive and negative square
are 6 and – 6.
roots of 36 b.  49 = 7
The positive square root of 49 is 7.
The negative square root of 4 is – 2. c.  4 – = 2

3. Evaluate the expression.
EXAMPLE 1
GUIDED PRACTICE
Find square roots
for Example 1
Evaluate the expression. –  1. 9 = 3
The negative square
is – 3.
roots of 9  2. 25 = 5
The positive square root of 25 is 5.
The positive and negative square
root of 64 as 8 and – 8.  64 3. – + = 8
The negative square
is – 9.
roots of 81  – 4. 81 = 9

4. Approximate a square root
EXAMPLE 2
Approximate a square root
FURNITURE
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 2 =
You need to find the side length s of the tabletop such that s This means that s is the positive square root of 945. You can use a table to determine whether 945 is a perfect square.

5. Approximate a square root Number 28 29 30 31 32
EXAMPLE 2
Approximate a square root
Number 28 29 30 31 32
Square of number
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
Write a compound inequality that compares 945 with both 900 and 961.
< 
961
900
945
Take positive square root of each number. 30 
945
< 31
Find square root of each perfect square.

6. Approximate a square root
EXAMPLE 2
Approximate a square root
Because 945 is closer to 961 than to 900, is closer to 31 than to 30. 
945
ANSWER
The side length of the tabletop is about 31 inches.

7. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
Approximate the square root to the nearest integer. 5.  32
You can use a table to determine whether 32 is a perfect square.
Square of number
Number 5 6 7 8 64 49 36 25
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.

8. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
25 < < 36
Write a compound inequality that compares 32 with both 25 and 36.
<  36 25 32
Take positive square root of each number. 5  32
< 6
Find square root of each perfect square.
Because 32 is closer to 36 than to 25, is closer to 6 than to 5.  32

9. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
Approximate the square root to the nearest integer. 6. 
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.

10. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
100< 103< 121
Write a compound inequality that compares 103 with both 100 and 121.
< 
121
100
103
Take positive square root of each number. 10 
103
< 11
Find square root of each perfect square.
Because 100 is closer to 103 than to 121, is closer to 10than to 11. 
103

11. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
Approximate the square root to the nearest integer. 7.  48 –
You can use a table to determine whether 48 is a perfect square.
Square of number
Number
– 6
– 7
– 8
– 9 81 64 49 36
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 49.

12. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
– 36 < – 48 < – 49
Write a compound inequality that compares 103 with both 100 and 121.
<  36 – 48 49
Take positive square root of each number.
– 6
< –7  48 –
Find square root of each perfect square.
Because 49is closer than to 36, – is closer to
– 7 than to – 6.  48

13. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
Approximate the square root to the nearest integer. 8. 
350 –
You can use a table to determine whether 48 is a perfect square.
Square of number
Number
– 17
– 18
– 19
– 20
400
361
324
187
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.

14. Approximate a square root for Example 2
GUIDED PRACTICE
Approximate a square root
for Example 2
– 324 < – 350 < – 361
Write a compound inequality that compares – 350 with both – 324 and – –361.
< 
324 –
350
361
Take positive square root of each number.
– 18
< 
350 –
<
– 19
Find square root of each perfect square.
Because 361 is closer than to 324, – is closer to
– 19 than to – 18. 
350

15. 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 81
100 No
Yes
Real Number?
Whole Number?
Integer?
Irrational Number?
Rational Number?
Number  24
100 81

16. EXAMPLE 4
Graph and order real numbers ,
Order the numbers from least to greatest: 4 3 –  5 13
–2.5 9 .
SOLUTION
Begin by graphing the numbers on a number line. 4 3
ANSWER
Read the numbers from left to right:
–2.5, –  5 9 13 . ,

17. EXAMPLE 4
GUIDED PRACTICE
Graph and order real numbers
for Examples 3 and 4
Tell whether each of the following numbers. A rational
number,an irrational number, an integer, or a whole
number: ,5.2, 0, , 4.1, There order the
number from least to greatest. 9.  7 9 2 – 20
SOLUTION
Begin by graphing the numbers on a number line. 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 9 –  20
4.4 =  7
2.6
5.2
4.1

18. Graph and order real numbers for Examples 3 and 4
GUIDED PRACTICE
Graph and order real numbers
for Examples 3 and 4 9 2 –
Read the numbers from left to right: ,  7 20
4.1
5.2.
Integer?
Number
Real Number?
Rational Number?
Irrational Number?
Whole Number? –  20
Yes No
Yes No No 9 2 –
Yes
Yes No No No No No No No No  7
Yes No
Yes No No
4.1
Yes No
Yes
Yes No
5.2
Yes No
Yes
Yes No

19. EXAMPLE 5
Rewrite a conditional statement in if-then form
Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample.
SOLUTION a.
Given: No integers are irrational numbers.
If-then form: If a number is an integer, then it is not an irrational number.
The statement is true.

20. EXAMPLE 5
Rewrite a conditional statement in if-then form b.
Given: All real numbers are rational numbers.
If-then form: If a number is a real number, then it is a rational number.
The statement is false. For example, is a real number but not a rational number.  2

21. EXAMPLE 5
GUIDED PRACTICE
Rewrite a conditional statement in if-then form
for Example 5
Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample.
All square roots of perfect squares are rational
number.
10.
SOLUTION
Given: All square roots of perfect squares are rational numbers.
If-then form: If a number is the square root of perfect square, then it is a irrational number.
The statement is true.

22. EXAMPLE 5
GUIDED PRACTICE
Rewrite a conditional statement in if-then form
for Example 5
Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample.
All repeating decimals are irrational number.
11.
SOLUTION
Given: All repeating decimals are rational numbers.
If-then form: If a number repeating decimals , then it is an irrational number.
The statement is false. For example, is a repeating decimals can be written as a rational number.

23. EXAMPLE 5
GUIDED PRACTICE
Rewrite a conditional statement in if-then form
for Example 5
Rewrite the given conditional statement in if-then form. Then tell whether the statement is true or false. If it is false, give a counterexample.
No integers are irrational number.
12.
SOLUTION
Given: No integers are irrational numbers.
If-then form: If a number is an integer, then it is not an irrational number
The statement is true.