2.7
Find Square Roots and Compare Real Numbers
Warm Up
Lesson Presentation
Lesson Quiz
2.7
Warm-Up
Find the square of the number.
1. 14
ANSWER
196
2. 16
ANSWER
256
Complete the statements using <, >, or =.
3. – 2.5 ? – 19
8
<
ANSWER
4.
5
21
?
6
25
ANSWER
<
2.7
Warm-Up
5. A square room has a side length of 25 feet. What is
its area?
ANSWER
625 ft2
2.7
Example 1
Evaluate the expression.
a.
+–  36
b.
 49
c.
– 4
= +
–6
The positive and negative square
roots of 36 are 6 and – 6.
=7
The positive square root of 49 is 7.
= –2
The negative square root of 4 is – 2.
2.7
Guided Practice
Evaluate the expression.
= –3
1.
– 9
2.
 25
3.
–+  64
= –+ 8
4.
–  81
= –9
=
5
2.7
Example 2
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
You need to find the side length s of the tabletop such
that s2 = 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.
2.7
Example 2
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
Write a compound inequality that
compares 945 with both 900 and 961.
Take positive square root of each
number.
Find square root of each perfect
square.
2.7
Example 2
Because 945 is closer to 961 than to 900,  945 is closer to
31 than to 30.
ANSWER
The side length of the tabletop is about 31 inches.
2.7
Guided Practice
Approximate the square root to the nearest integer.
5.  32
6
6.  103
10
7. –  48
–7
8. –  350
– 19
2.7
Example 3
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 .
Irrational
Whole
Number? Integer? Number?
Real
Number?
Rational
Number?
 24
Yes
No
Yes
No
No
 100
Yes
Yes
No
Yes
Yes
–  81
Yes
Yes
No
Yes
No
Number
2.7
Guided Practice
9
–
9. Tell whether
is a real number, a rational number,
2
an irrational number, an integer, or a whole number.
ANSWER
real number and rational number
2.7
Example 4
Compare
4
and  13 .
3
SOLUTION
Graph the numbers on a number line.
ANSWER
Because
4
3
is to the left of  13,.
4
3
<  13 .
2.7
Guided Practice
10. Copy and complete using < or >:
(a)
– 5
–2.5
?
ANSWER
>
(b)
7
?
ANSWER
<
14
5
2.7
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.
SOLUTION
a.
Given: No fractions are irrational numbers.
If-then form: If a number is a fraction, then it is not an
irrational number.
The statement is true.
2.7
b.
Example 5
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,  2 is a real
number but not a rational number.
2.7
Guided Practice
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.
11.
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 rational number.
The statement is true.
2.7
Guided Practice
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.
12.
All repeating decimals are irrational numbers.
If-then form: If a number is a repeating decimal, then
it is an irrational number.
The statement is false. For example, 0.333… is a
repeating decimal and can be written as 1 , so it is
3
a rational number.
2.7
Guided Practice
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.
13.
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.
2.7
Lesson Quiz
Evaluate the expression.
1. +–  289
ANSWER
2.
+– 17
–  36
ANSWER
–6
Approximate the square root to the nearest integer.
3.
–  21
ANSWER
4.
–5
 620
ANSWER
25
2.7
5.
Lesson Quiz
A square courtyard has an area of 272 square feet.
What is the side length of the courtyard to the
nearest foot?
ANSWER
16 ft