Transcript Lesson 2.7 Finding Square Roots and Compare Real Numbers

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.
EXAMPLE 1
Find square roots
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.
EXAMPLE
1
for Example
1
Find square
roots
GUIDED PRACTICE
Evaluate the expression.
1.
2.
– 9
 25
= –3
=
The negative square
roots of 9 is – 3.
5
The positive square root of 25 is 5.
The positive and negative square
root of 64 as 8 and – 8.
The negative square
roots of 81 is – 9.
3.
–+  64
= –+ 8
4.
–  81
= –9
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
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.
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.
 900 < 945 <  961
Take positive square root of each
number.
30 < 945 < 31
Find square root of each perfect
square.
EXAMPLE 2
Approximate a square root
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.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
Approximate the square root to the nearest integer.
5.  32
You can use a table to determine whether 32 is a
perfect square.
Number
Square of
number
5
6
7
8
25
36
49
64
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.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
25 < 32 < 36
Write a compound inequality that
compares 32 with both 25 and 36.
 25 < 32 <  36
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,  32 is closer to 6
than to 5.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
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.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
100< 103< 121
Write a compound inequality that
compares 103 with both 100 and 121.
 100 < 103 <  121
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,  103 is closer to
10than to 11.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
Approximate the square root to the nearest integer.
7. –  48
You can use a table to determine whether 48 is a
perfect square.
Number
–6
–7
–8
–9
Square of
number
36
49
64
81
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.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
– 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 < –  48 < –7
Find square root of each perfect
square.
Because 49is closer than to 36, –  48 is closer to
– 7 than to – 6.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
Approximate the square root to the nearest integer.
8. –  350
You can use a table to determine whether 48 is a
perfect square.
Number
– 17
– 18
– 19
– 20
Square of
number
187
324
361
400
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.
EXAMPLE
2
for Example
Approximate
a square 2root
GUIDED PRACTICE
– 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, –  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 .
Number
Real
Number?
Rational
Number?
Irrational
Whole
Number? Integer? Number?
 24
Yes
No
Yes
No
No
 100
Yes
Yes
No
Yes
Yes
 81
Yes
Yes
No
Yes
No
EXAMPLE 4
Graph and order real numbers
Order the numbers from least to greatest: 4 ,–  5 ,  13 ,
3
–2.5 , 9 .
SOLUTION
Begin by graphing the numbers on a number line.
ANSWER
Read the numbers from left to right:
–2.5, –  5 , 4 , 9 , 13 .
3
EXAMPLE
4
fororder
Examples
3 and
Graph and
real numbers
GUIDED PRACTICE
4
9. Tell whether each of the following numbers. A rational
number,an irrational number, an integer, or a whole
–  20 . There order the
number: – 9 ,5.2, 0, 7 , 4.1,
2
number from least to greatest.
SOLUTION
Begin by graphing the numbers on a number line.
9  20 = 4.4
–
2
0
 7 = 2.6
–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5
4.1
5.2
EXAMPLE
4
fororder
Examples
3 and
Graph and
real numbers
GUIDED PRACTICE
4
Read the numbers from left to right:
9
–  20 ,– , 0 ,  7 , 4.1 , 5.2.
2
Number
Real
Number?
Rational
Number?
Irrational
Number?
Integer?
Whole
Number?
–  20
Yes
No
Yes
No
No
9
2
Yes
Yes
No
No
No
0
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
–
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.
EXAMPLE 5
b.
Rewrite a conditional statement in if-then form
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.
EXAMPLE
5
Example 5statement in if-then form
Rewrite afor
conditional
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.
10. All square roots of perfect squares are rational
number.
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.
EXAMPLE
5
Example 5statement in if-then form
Rewrite afor
conditional
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 repeating decimals are irrational number.
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, 0.333 is a
repeating decimals can be written as a rational
number.
EXAMPLE
5
Example 5statement in if-then form
Rewrite afor
conditional
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. No integers are irrational number.
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.