Transcript Square Roots/Rational Numbers - St. Croix Central School

Square Roots/Rational
Numbers
1-5 Square Roots and Real Numbers
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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
8. –1
5
6
–1.83
Objectives
Evaluate expressions containing square roots.
Classify numbers within the real number
system.
Vocabulary
square root
perfect square
real numbers
natural numbers
whole numbers
integers
rational numbers
terminating decimal
repeating decimal
irrational 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
–
=4
Positive square
root of 16
= –4
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
02
12
22 32
16 25 36 49 64 81 100
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.
Example 1: Finding Square Roots of
Perfect Squares
Find each square root.
A.
42 = 16
=4
B.
32 = 9
= –3
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.
Example 1C: Finding Square Roots of
Perfect Squares
Find the square root.
Think: What number squared
equals 25 ?
81
Positive square root
positive
5
.
9
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
negative 5.
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.
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.
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.
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
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.
4
 3.6
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.
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.
All numbers that can be represented on a
number line are called real numbers and can
be classified according to their characteristics.
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
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: ,
,
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
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
Lesson Quiz
Find each square root.
-8
2.
1.
12
3.
3
7
4.
–
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.
6. 1 rational, integer, whole number, natural
number, terminating decimal
7. –3.89 rational, repeating decimal
8.
irrational
1
2