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Square Roots and Real Numbers

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz

2

5

5

9

5

6

3

8

5

–1

Warm Up

Simplify each expression.

1. 62

121

2. 112

36

25

36

81

4.

3. (–9)(–9)

Write each fraction as a decimal.

0.4

5.

6.

0.5

–1.83

7.

5.375

8.

Objectives

Evaluate expressions containing square roots.

Classify numbers within the real number system.

Vocabulary

square rootterminating decimal

perfect square repeating decimal

real numbersirrational numbers

natural numbers

whole numbers

integers

rational 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.

Positive square

root of 16

=4

4 4 = 42= 16

= –4

–

(–4)(–4) = (–4)2= 16

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

16

25

36

49

64

81

100

02

12

22

32

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.

A.

= 4

B.

= –3

Example 1: Finding Square Roots of

Perfect Squares

Find each square root.

Think: What number squared equals 16?

42 = 16

Positive square root positive 4.

Think: What is the opposite of the

square root of 9?

32 = 9

Negative square root negative 3.

25

81

Think: What number squared equals ?

5

9

Positive square root positive .

Example 1C: Finding Square Roots of

Perfect Squares

Find the square root.

1a.

Positive square root positive 2.

1b.

= 2

Negative square root negative 5.

Check It Out! Example 1

Find the square root.

Think: What number squared

equals 4?

22 = 4

52 = 25

Think: What is the opposite of the square root of 25?

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.

1

Understand the problem

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?

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.

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.

2

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

is greater than 3.6.

is less than 3.7.

Because 13 is closer to 12.96 than to 13.69, is closer to 3.6 than to 3.7.

3.6

Example 2 Continued

Guess 3.6: 3.62 = 12.96 too low

Guess 3.7: 3.72 = 13.69 too high

3.6

3.7

4

3

4

Look Back

Example 2 Continued

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.

Use a guess and check method to estimate .

is greater than 6.1.

is less than 6.2.

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?

Guess 6.1 6.12 = 37.21 too low

Guess 6.2 6.22 = 38.44 too high

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.

Rational numbers can be expressed in the form ,

where a and b are both integers and b ≠ 0: , , .

1

2

a

b

7

1

9

10

Natural numbersare 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, …

Repeating decimalsare 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 . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: ,

,

a

b

Terminating decimals are rational numbers in

decimal form that have a finite number of digits:

1.5, 2.75, 4.0

Example 3: Classifying Real Numbers

Write all classifications that apply to each

Real number.

A. –32

32 can be written as a fraction and a decimal.

32

1

–32 = – = –32.0

rational number, integer, terminating decimal

B. 5

5 can be written as a fraction and a decimal.

5

1

5 = = 5.0

rational number, integer, whole number, natural

number, terminating decimal

7 can be written as a repeating decimal.

4

9

3a. 7

67 9 = 7.444… = 7.4

49

12 1

–12 = – = –12.0

3c.

= 3.16227766…

Check It Out! Example 3

Write all classifications that apply to each real number.

rational number, repeating decimal

3b. –12

32 can be written as a fraction and a decimal.

rational number, terminating decimal, integer

The digits continue with no pattern.

irrational number

7. –3.89

Lesson Quiz

Find each square root.

1

2

3

7

-8

3.

4.

–

1.

12

2.

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.

rational, integer, whole number, natural number, terminating decimal

6. 1

rational, repeating decimal

8.

irrational