Transcript 1-5 Roots and Irrational Numbers

1-5
Roots and Irrational Numbers
Preview
Warm Up
California Standards
Lesson Presentation
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Roots and Irrational Numbers
Warm Up
Simplify each expression.
2. 112 121
1. 62 36
3. (–9)(–9) 81
4.
25
36
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
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Roots and Irrational Numbers
California
Standards
2.0 Students understand and use such
operations as taking the opposite, finding the
reciprocal, taking a root, and raising to a
fractional power. They understand and use the
rules of exponents.
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Roots and Irrational Numbers
Vocabulary
square root
principal square root
perfect square
cube root
natural numbers
whole numbers
integers
rational numbers
terminating decimal
repeating decimal
irrational numbers
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A number that is multiplied by itself to form a
product is a square root of that product. The
radical symbol
is used to represent square
roots. For nonnegative numbers, the operations
of squaring and finding a square root are inverse
operations. In other words, for x ≥ 0,
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
Roots and Irrational Numbers
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The principal square root of a number is the
positive square root and is represented by . A
negative square root is represented by – . The
symbol
is used to represent both square roots.
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
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Writing Math
The small number to the left of the root is the
index. In a square root, the index is understood
to be 2. In other words,
is the same as
.
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A number that is raised to the third power to form
a product is a cube root of that product. The
symbol
indicates a cube root. Since 23 = 8,
= 2. Similarly, the symbol
indicates a fourth
root: 2 = 16, so
= 2.
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Additional Example 1: Finding Roots
Find each root.
Think: What number squared equals 81?
Think: What number squared equals 25?
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Additional Example 1: Finding Roots
Find the root.
C.
Think: What number cubed equals
–216?
= –6
(–6)(–6)(–6) = 36(–6) = –216
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Check It Out! Example 1
Find each root.
a.
Think: What number squared
equals 4?
b.
Think: What number squared
equals 25?
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Check It Out! Example 1
Find the root.
c.
Think: What number to the fourth
power equals 81?
Roots and Irrational Numbers
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Additional Example 2: Finding Roots of Fractions
Find the root.
A.
Think: What number squared
equals
Roots and Irrational Numbers
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Additional Example 2: Finding Roots of Fractions
Find the root.
B.
Think: What number cubed equals
Roots and Irrational Numbers
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Additional Example 2: Finding Roots of Fractions
Find the root.
C.
Think: What number squared
equals
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Check It Out! Example 2
Find the root.
a.
Think: What number squared
equals
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Roots and Irrational Numbers
Check It Out! Example 2
Find the root.
b.
Think: What number cubed
equals
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Check It Out! Example 2c
Find the root.
c.
Think: What number squared
equals
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Square roots of numbers that are not perfect
squares, such as 15, 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.
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Additional Example 3: Art Application
As part of her art project, Shonda will need to
make a paper square covered in glitter. Her
tube of glitter covers 13 in². Estimate to the
nearest tenth the side length of a square with
an area of 13 in².
Since the area of the square is 13 in², then each
side of the square is
in. 13 is not a perfect
square, so find two consecutive perfect squares
that
is between: 9 and 16.
is between
and
, or 3 and 4. Refine the estimate.
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Additional Example 3 Continued
3.5
3.52 = 12.25 too low
3.6
3.62 = 12.96 too low
3.65 3.652 = 13.32 too high
Since 3.6 is too low and 3.65 is too high,
is
between 3.6 and 3.65. Round to the nearest tenth.
The side length of the paper square is
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Writing Math
The symbol ≈ means “is approximately equal to.”
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Check It Out! Example 3
What if…? Nancy decides to buy more wildflower
seeds and now has enough to cover 26 ft2.
Estimate to the nearest tenth the side length of a
square garden with an area of 26 ft2.
Since the area of the square is 26 ft², then each
side of the square is
ft. 26 is not a perfect
square, so find two consecutive perfect squares
that
is between: 25 and 36.
is between
and
, or 5 and 6. Refine the estimate.
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Check It Out! Example 3 Continued
5.0
5.02 = 25 too low
5.1
5.12 = 26.01 too high
Since 5.0 is too low and 5.1 is too high,
is
between 5.0 and 5.1. Rounded to the nearest tenth,
 5.1.
The side length of the square garden is
 5.1 ft.
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Real numbers 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 the whole numbers and their
opposites: –3, –2, –1, 0, 1, 2, 3, …
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Rational numbers are numbers that can be
expressed in the form , where a and b are both
integers and b ≠ 0. When expressed as a
decimal, a rational number is either a terminating
decimal or a repeating decimal.
• A terminating decimal has a finite number of
digits after the decimal point (for example, 1.25,
2.75, and 4.0).
• A repeating decimal has a block of one or more
digits after the decimal point that repeat
continuously (where all digits are not zeros).
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Irrational numbers are all numbers that are not
rational. They cannot be expressed in the form
where a and b are both integers and b ≠ 0. They
are neither terminating decimals nor repeating
decimals. For example:
0.10100100010000100000…
After the decimal point, this number contains 1
followed by one 0, and then 1 followed by two
0’s, and then 1 followed by three 0’s, and so on.
This decimal neither terminates nor repeats, so it is
an irrational number.
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If a whole number is not a perfect square, then its
square root is irrational. For example, 2 is not a
perfect square and
is irrational.
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The real numbers are made up of all rational
and irrational numbers.
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Reading Math
Note the symbols for the sets of numbers.
R: real numbers
Q: rational numbers
Z: integers
W: whole numbers
N: natural numbers
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Additional Example 4: Classifying Real Numbers
Write all classifications that apply to each
real number.
A. –32
32
1
–32 = –32.0
–32 = –
–32 can be written in the form
.
–32 can be written as a terminating
decimal.
rational number, integer, terminating decimal
B.
irrational
14 is not a perfect square, so
irrational.
is
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Check It Out! Example 4
Write all classifications that apply to each real
number.
a. 7
7 4 can be written in the form
9
.
can be written as a repeating
decimal.
rational number, repeating decimal
b. –12
–12 can be written in the form .
–12 can be written as a
terminating decimal.
rational number, terminating decimal, integer
67  9 = 7.444… = 7.4
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Check It Out! Example 4
Write all classifications that apply to each real
number.
irrational
10 is not a perfect square, so
is irrational.
100 is a perfect square, so
is rational.
10 can be written in the form
and as a terminating decimal.
natural, rational, terminating decimal, whole, integer
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Lesson Quiz
Find each square root.
1.
3
2.
3.
5
4.
1
5. The area of a square piece of cloth is 68 in2.
Estimate to the nearest tenth the side length
of the cloth.  8.2 in.
Write all classifications that apply to each
real number.
6. –3.89 rational, repeating
decimal
7.
irrational