Transcript Square Roots and Irrational Numbers

PRE-ALGEBRA
Lesson 11-1 Warm-Up
PRE-ALGEBRA
Square Roots and Irrational
Numbers (11-1)
What is a
“perfect
square”?
perfect square: a number times itself - It’s called this, because the area of a
square is a number times itself (one of its sides times itself, or one side squared).
Examples:
The first 15 perfect squares are:
12 22 32 42 52 62 72 82 92 102 112 122 132 142 152
        


  

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
What is a
“square
root”??
square root: one side of a perfect square - the inverse, or opposite, of squaring a
number
Example: 2 and -2 are the square roots of 4, since 22 = 4 and (-2)2 = 4.
Note: Every number has a positive and negative square root (since a negative
times a negative is a positive).
Example:
PRE-ALGEBRA
Square Roots and Irrational
Numbers (11-1)
What are the
parts of “square
root”?
A radical symbol
indicate a square root. The number inside the
radical sign is called the radicand. You can indicate that a number can be
positive or negative with a , which means “plus or minus”. The square root of a
negative number is undefined (impossible) meaning you can’t have a negative
number inside a radical sign. However, the negative can be outside the radical.
This is called a negative square root.(Example: - 16 means “the negative
square root of 16.”)
How do you find
the square root
of a number?
To find the root of a number, think of a number that equals the number when it is
multiplied times itself
Example: Find the two square roots of
81 .
9 • 9 = 81
Find a number that equals 81 when
(-9) • (-9) = 81
multiplied by itself.
The square roots of 81 are 9 and -9.
Example: Simplify 144 .
12 x 12 = 144
Find a number that equals 144 when
(-12) x (-12) = 144
multiplied by itself.
The square roots of 144 are 12 and -12.
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
Simplify each expression.
a.
b.
c. –
d.
e.
The square roots are 5 and -5.
25 = ± 5
3
9
=± 5
25
The square roots are
64 = –8
The square roots are -8 and 8.
–49 is undefined
1
=
16
3
3
and – .
5
5
± 1
4
For real numbers, the square root of a negative
number is undefined.
1
The square root is ± .
4
.
PRE-ALGEBRA
Square Roots and Irrational
Numbers (11-1)
How do you
A radical is a grouping sign, so when you simplify an expression using the order
simplify an
of operations (PEDMAS), radicals are the same as the “P” for parenthesis. In
expression with a other words, simplify under the radical sign (radicand) first.
root?
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
a. Simplify 3 + 2 • √ 50 – 14
3 + 2 • √ 50 – 14
= 3 + 2 • √36
Work inside grouping symbols.
=3+2•6
Simplify the root.
= 3 + 12
Multiply.
= 15
Add.
PRE-ALGEBRA
Square Roots and Irrational
Numbers (11-1)
How can you
estimate the
square root of a
number?
You can estimate the square root of a number by figuring out what perfect
squares its between.
Examples: Between what two consecutive integers is
14.52 .
14.52 is between 3 and 4.
Examples: Estimate
8 .
Place
8 on a number line between
its perfect squares.
8 is between the perfect squares of
4 and
9 . Since
to
9 than 4,
8 is closer to 3 than it is to 2.
8  3
8 is closer
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
b. A square tile has an area of 170 square inches.
About how long is each side of the tile?
Since 170 is not a perfect square, find the perfect square closest to
170. 132 = 169 and 14 2 = 196, so √170 is between √169 and √196.
Because 170 is closer to 169 than to 196,
√170 is closer to 13 than to 14.
Each side of the tile is about 13 inches long.
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
c. You can use the formula d = 1.5h to estimate the
distance d, in miles, to a horizon line when your eyes are h feet
above the ground. Estimate the distance to the horizon seen by
a lifeguard whose eyes are 20 feet above the ground.
d=
1.5h
Use the formula.
d=
1.5(20)
Replace h with 20.
d=
30
Multiply.
25 <
25 = 5
30 <
36
Find perfect squares close to 30.
Find the square root of the
closest perfect square.
The lifeguard can see about 5 miles to the horizon.
PRE-ALGEBRA
Square Roots and Irrational
Numbers (11-1)
What are “natural
numbers”, “whole
numbers”, and
“integers”?
natural numbers: the positive numbers from 1 to infinity (1, 2,3,…….)
whole numbers: the positive numbers from 0 to infinity (0,1,2,3…….)
Integers: all of the positive and negative numbers (…-2, -1, 0, 1, 2,…)
What are “rational” and
“irrational” numbers?
rational numbers: any number that can be expressed as a fraction or
whose decimal form either terminates or repeats (examples: 6, or 6.0,
terminates, or ends, and 8.242424… repeats)
irrational numbers: any number that cannot be expressed as a fraction
or whose decimal form doesn’t terminates or repeats (examples:
0.101001000… , , and  3 do not terminate or repeat)
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
Identify each number as rational or irrational. Explain.
a.
49
rational, because 49 is a perfect square
b. 0.16
rational, because it is a terminating decimal
c. 3
irrational, because 3 is not a perfect square
d. 0.3333 . . .
rational, because it is a repeating decimal
e. – 15
irrational, because 15 is not a perfect square
f. 12.69
rational, because it is a terminating decimal
g. 0.1234567 . . .
irrational, because it neither terminates nor repeats
PRE-ALGEBRA
Square Roots and Irrational Numbers
LESSON 11-1
Lesson Quiz
Simplify each square root or estimate to the nearest integer.
1. –
100
2.
–10
57
8
Identify each number as rational or irrational.
3.
48
irrational
4. 0.0125
rational
5. The formula d = 1.5h , where h equals the height, in feet, of the
viewer’s eyes, estimates the distance d, in miles, to the horizon from
the viewer. Find the distance to the horizon for a person whose eyes
are 6 ft above the ground.
3 mi
PRE-ALGEBRA