1-5. Square Roots and Real Numbers. Holt Algebra 1. Warm Up. Lesson Presentation. Lesson Quiz. 2 5. 5 9. 5 6. 3 8. 5. –1. Warm Up Simplify each expression. 1. 6 2. 121. 2. 11 2. 36. 25 36. 81. 4. 3. (–9)( – 9). Write each fraction as a decimal. 0.4. 5. 6. 0.5.
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1-5
Square Roots and Real Numbers
Holt Algebra 1
Warm Up
Lesson Presentation
Lesson Quiz
2
5
5
9
5
6
3
8
5
–1
Warm Up
Simplify each expression.
1. 62
121
2. 112
36
25
36
81
4.
3. (–9)(–9)
Write each fraction as a decimal.
0.4
5.
6.
0.5
–1.83
7.
5.375
8.
Objectives
Evaluate expressions containing square roots.
Classify numbers within the real number system.
Vocabulary
square rootterminating decimal
perfect square repeating decimal
real numbersirrational numbers
natural numbers
whole numbers
integers
rational numbers
A number that is multiplied by itself to form a
product is called a square root of that product.
The operations of squaring and finding a square
root are inverse operations.
The radical symbol , is used to represent square roots. Positive real numbers have two
square roots.
Positive square
root of 16
=4
4 4 = 42= 16
= –4
–
(–4)(–4) = (–4)2= 16
Negative square
root of 16
The nonnegative square root is represented by . The negative square root is represented by – .
A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table.
0
1
4
9
16
25
36
49
64
81
100
02
12
22
32
42
52
62
72
82
92
102
Reading Math
The expression does not represent
a real number because there is no real number that can be multiplied by itself to form a product of –36.
A.
= 4
B.
= –3
Example 1: Finding Square Roots of
Perfect Squares
Find each square root.
Think: What number squared equals 16?
42 = 16
Positive square root positive 4.
Think: What is the opposite of the
square root of 9?
32 = 9
Negative square root negative 3.
25
81
Think: What number squared equals ?
5
9
Positive square root positive .
Example 1C: Finding Square Roots of
Perfect Squares
Find the square root.
1a.
Positive square root positive 2.
1b.
= 2
Negative square root negative 5.
Check It Out! Example 1
Find the square root.
Think: What number squared
equals 4?
22 = 4
52 = 25
Think: What is the opposite of the square root of 25?
The square roots of many numbers like , are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.
1
Understand the problem
Example 2: Problem-Solving Application
As part of her art project, Shonda will
need to make a square covered in glitter.
Her tube of glitter covers 13 square
inches. What is the greatest side length
Shonda’s square can have?
The answer will be the side length of the
square.
List the important information:
• The tube of glitter can cover an area of 13 square inches.
Make a Plan
The side length of the square is because
=
13.
Because 13 is not a perfect
square, is not a whole number. Estimate
to the nearest tenth.
Find the two whole numbers that is
between. Because 13 is between the perfect
squares 0 and 16. is between and
, or between 3 and 4.
2
Example 2 Continued
Because 13 is closer to 16 than to 9,
is closer to 4 than to 3.
3
4
You can use a guess-and-check
method to estimate .
Example 2 Continued
3
Solve
is greater than 3.6.
is less than 3.7.
Because 13 is closer to 12.96 than to 13.69, is closer to 3.6 than to 3.7.
3.6
Example 2 Continued
Guess 3.6: 3.62 = 12.96 too low
Guess 3.7: 3.72 = 13.69 too high
3.6
3.7
4
3
4
Look Back
Example 2 Continued
A square with a side length of 3.6 inches
would have an area of 12.96 square inches.
Because 12.96 is close to 13, 3.6 inches
is a reasonable estimate.
Use a guess and check method to estimate .
is greater than 6.1.
is less than 6.2.
Check It Out! Example 2
What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 38 ft2. What is the side length of a square garden with an area of 38 ft2?
Guess 6.1 6.12 = 37.21 too low
Guess 6.2 6.22 = 38.44 too high
A square garden with a side length of 6.2 ft would have an area of 38.44 ft2. 38.44 ft is close to 38, so 6.2 is a reasonable answer.
All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.
Rational numbers can be expressed in the form ,
where a and b are both integers and b ≠ 0: , , .
1
2
a
b
7
1
9
10
Natural numbersare the counting numbers: 1, 2, 3, …
Whole numbers are the natural numbers and zero:
0, 1, 2, 3, …
Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, …
Repeating decimalsare rational numbers in
decimal form that have a block of one or more
digits that repeat continuously: 1.3, 0.6, 2.14
Irrational numbers cannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: ,
,
a
b
Terminating decimals are rational numbers in
decimal form that have a finite number of digits:
1.5, 2.75, 4.0
Example 3: Classifying Real Numbers
Write all classifications that apply to each
Real number.
A. –32
32 can be written as a fraction and a decimal.
32
1
–32 = – = –32.0
rational number, integer, terminating decimal
B. 5
5 can be written as a fraction and a decimal.
5
1
5 = = 5.0
rational number, integer, whole number, natural
number, terminating decimal
7 can be written as a repeating decimal.
4
9
3a. 7
67 9 = 7.444… = 7.4
49
12 1
–12 = – = –12.0
3c.
= 3.16227766…
Check It Out! Example 3
Write all classifications that apply to each real number.
rational number, repeating decimal
3b. –12
32 can be written as a fraction and a decimal.
rational number, terminating decimal, integer
The digits continue with no pattern.
irrational number
7. –3.89
Lesson Quiz
Find each square root.
1
2
3
7
-8
3.
4.
–
1.
12
2.
5. The area of a square piece of cloth is 68 in2.
How long is each side of the piece of cloth?
Round your answer to the nearest tenth of an
inch.
8.2 in.
Write all classifications that apply to each real number.
rational, integer, whole number, natural number, terminating decimal
6. 1
rational, repeating decimal
8.
irrational