Transcript square root

1-5 Square Roots and Real Numbers
Warm Up
Simplify each expression.
2. 112 121
1. 62 36
3. (–9)(–9) 81
25
36
4.
Write each fraction as a decimal.
5. 2 0.4
6. 5
0.5
5
9
7. 5 3 5.375
8
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8. –1
5
6
–1.83
1-5 Square Roots and Real Numbers
Objectives
Evaluate expressions containing square roots.
Classify numbers within the real number
system.
Holt Algebra 1
1-5 Square Roots and Real 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.
4  4 = 42 = 16
(–4)(–4) = (–4)2 = 16
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–
=4
Positive square
root of 16
= –4
Negative square
root of 16
1-5 Square Roots and Real Numbers
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
02
12
22 32
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9
16 25 36 49 64 81 100
42 52
62
72
82
92 102
1-5 Square Roots and Real Numbers
Example 1: Finding Square Roots of
Perfect Squares
Find each square root.
A.
42 = 16
=4
B.
32 = 9
= –3
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Think: What number squared equals 16?
Positive square root
positive 4.
Think: What is the opposite of the
square root of 9?
Negative square root
negative 3.
1-5 Square Roots and Real Numbers
Example 1C: Finding Square Roots of
Perfect Squares
Find the square root.
Think: What number squared
equals 25 ?
81
Positive square root
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positive
5
.
9
1-5 Square Roots and Real Numbers
Check It Out! Example 1
Find the square root.
1a.
22 = 4
=2
Think: What number squared
equals 4?
Positive square root
positive 2.
1b.
52 = 25
Think: What is the opposite of the
square root of 25?
Negative square root
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negative 5.
1-5 Square Roots and Real Numbers
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?
1
Understand the problem
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.
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1-5 Square Roots and Real Numbers
Example 2 Continued
2
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.
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1-5 Square Roots and Real Numbers
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
.
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1-5 Square Roots and Real Numbers
Example 2 Continued
3
Solve
Guess 3.6: 3.62 = 12.96
too low
Guess 3.7:
too high
3
3.72 = 13.69
3.6
is greater than 3.6.
is less than 3.7.
3.7
Because 13 is closer to 12.96 than to
13.69,
is closer to 3.6 than to 3.7.
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4
 3.6
1-5 Square Roots and Real Numbers
Example 2 Continued
4
Look Back
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.
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1-5 Square Roots and Real Numbers
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?
Use a guess and check method to estimate
Guess 6.1
6.12 = 37.21 too low
Guess 6.2
6.22 = 38.44 too high
is greater than 6.1.
is less than 6.2.
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.
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.
1-5 Square Roots and Real Numbers
All numbers that can be represented on a
number line are called real numbers and can
be classified according to their characteristics.
Holt Algebra 1
1-5 Square Roots and Real Numbers
Natural numbers are 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, …
Rational numbers can be expressed in the form a ,
b
where a and b are both integers and b ≠ 0:
1, 7, 9 .
2 1 10
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1-5 Square Roots and Real Numbers
Terminating decimals are rational numbers in
decimal form that have a finite number of digits:
1.5, 2.75, 4.0
Repeating decimals are 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 a . They include square roots of whole
b
numbers that are not perfect squares and
nonterminating decimals that do not repeat: ,
,
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1-5 Square Roots and Real Numbers
Example 3: Classifying Real Numbers
Write all classifications that apply to each
Real number.
A. –32
32
–32 = –
= –32.0
1
32 can be written as a
fraction and a decimal.
rational number, integer, terminating decimal
B. 5
5 can be written as a
5
5 = = 5.0
fraction and a decimal.
1
rational number, integer, whole number, natural
number, terminating decimal
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1-5 Square Roots and Real Numbers
Check It Out! Example 3
Write all classifications that apply to each real
number.
4
4
7 9 can be written as a
3a. 7
repeating decimal.
9
67  9 = 7.444… = 7.4
rational number, repeating decimal
3b. –12
32 can be written as a
12
–12 = – 1 = –12.0 fraction and a decimal.
rational number, terminating decimal, integer
3c.
= 3.16227766… The digits continue with no
pattern.
irrational number
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1-5 Square Roots and Real Numbers
Homework
Pg. 35 12-30 evens,
31-34, 48-62 evens
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