2. Lesson 11-1 Warm-Up

3. Square Roots and Irrational
Numbers (11-1)
What is a “perfect square”?
What is a “square root”??
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:
              
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
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:

4. Square Roots and Irrational
Numbers (11-1)
What are the parts of “square root”? How do you find the square root of a number?
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.”) 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 • 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 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.

5. Simplify each expression.
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
Simplify each expression. a
The square roots are 5 and -5.
= ± 5 b. 9 25 3 5
The square roots are and – .
= ±
c. – 64
The square roots are -8 and 8.
= –8
d. –49
For real numbers, the square root of a negative
number is undefined.
is undefined 1 16 1 4
The square root is . = e.
± .

6. Square Roots and Irrational
Numbers (11-1)
How do you simplify an expression with a root?
A radical is a grouping sign, so when you simplify an expression using the order of operations (PEDMAS), radicals are the same as the “P” for parenthesis. In other words, simplify under the radical sign (radicand) first.

7. Work inside grouping symbols.
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
a. Simplify • √ 50 – 14
3 + 2 • √ 50 – 14
= • √36
Work inside grouping symbols.
= • 6
Simplify the root. =
Multiply.
= 15
Add.

8. 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 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 8 is closer to 9 than 4, 8 is closer to 3 than it is to 2. 8  3

9. b. A square tile has an area of 170 square inches.
Square Roots and Irrational Numbers
Square Roots
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? 2
Since 170 is not a perfect square, find the perfect square closest to = 169 and 14 = 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.

10. Find perfect squares close to 30. 25 30 36 <
Square Roots and Irrational Numbers
LESSON 11-1
Additional Examples
c. You can use the formula d = h 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.
Use the formula.
d = h
Replace h with 20.
d = (20)
Multiply.
d = 30
Find perfect squares close to 30. 25 30 36
<
Find the square root of the
closest perfect square.
25 = 5
The lifeguard can see about 5 miles to the horizon.

11. Square Roots and Irrational
Numbers (11-1)
What are “natural numbers”, “whole numbers”, and “integers”?
What are “rational” and “irrational” numbers?
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,…)
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 … repeats)
irrational numbers: any number that cannot be expressed as a fraction or whose decimal form doesn’t terminates or repeats (examples: … , , and  3 do not terminate or repeat)

12. Identify each number as rational or irrational. Explain.
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. 
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. 
irrational, because it neither terminates nor repeats

13. Simplify each square root or estimate to the nearest integer.
Square Roots and Irrational Numbers
LESSON 11-1
Lesson Quiz
Simplify each square root or estimate to the nearest integer.
1. –
Identify each number as rational or irrational.
5. The formula d = h , 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.
–10 8
irrational
rational
3 mi