Transcript Square Root

2.7 Square Roots &
Comparing Real Numbers



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
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.
GUIDED PRACTICE
Evaluate the expression.
1.
– 9
= –3
The negative square root of 9 is -3
The positive square root of 25 is 5.
2.
 25
3.
–+  64
= –+ 8
The positive and negative square
root of 64 as 8 and – 8.
4.
–  81
= –9
The negative square root of 81 is -9
=
5
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
2
s = 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
 900 < 945 <  961
30 < 945 < 31
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.
GUIDED PRACTICE
Approximate the square root to the nearest integer.
1.  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.
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.
GUIDED PRACTICE
Approximate the square root to the nearest integer.
2.  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.
GUIDED PRACTICE
100< 103< 121
 100 < 103 <  121
10 < 103 < 11
Write a compound inequality that
compares 103 with both 100 and 121.
Take positive square root of each
number.
Find square root of each perfect square.
Because 100 is closer to 103 than to 121,  103 is closer to 10
than to 11.
GUIDED PRACTICE
Approximate the square root to the nearest integer.
3. –  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.
GUIDED PRACTICE
– 36 < – 48 < – 49
–  36 < –  48 < –  49
– 6 < –  48 < –7
Write a compound inequality that
compares 103 with both 100 and 121.
Take positive square root of each
number.
Find square root of each perfect square.
Because 49is closer than to 36, –  48 is closer to
– 7 than to – 6.
GUIDED PRACTICE
Approximate the square root to the nearest integer.
4. –  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.
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
Yes
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
GUIDED PRACTICE
Order the numbers from least to greatest:
9
 ,5.2,  20, 7, 4.1, 0
2
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
9
Read the numbers from left to right:  ,  20,0, 7,4.1,5.2
2
4.1
5.2
13
95
GUIDED PRACTICE
•Classify the following numbers as Real, Rational, Irrational, Integer and/or Whole:
4
2.5,  5, , 9, 13
3
Real? Rational? Irrational? Integer? Whole?
-2.5
5
4/3
9
13