Square Roots/Rational Numbers

1-5 Square Roots and Real Numbers

Holt Algebra 1

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Warm UpWarm Up

Warm UpSimplify each expression.

1. 62 36 2. 112 121

3. (–9)(–9) 81 4. 2536

Write each fraction as a decimal.

5. 25

596.

7. 5 38

8. –1 56

0.4

5.375

0.5

–1.83

Evaluate expressions containing square roots.

Classify numbers within the real number system.

Objectives

square root terminating decimalperfect square repeating decimalreal numbers irrational numbersnatural numberswhole numbersintegersrational numbers

Vocabulary

A number that is multiplied by itself to form aproduct is called a square root of that product.The operations of squaring and finding a squareroot are inverse operations.

The radical symbol , is used to represent square roots. Positive real numbers have twosquare roots.

4 4 = 42 = 16 = 4 Positive squareroot of 16

(–4)(–4) = (–4)2 = 16 = –4 Negative squareroot of 16

–

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

02

1

12

1004

22

9

32

16

42

25

52

36

62

49

72

64

82

81

92 102

The nonnegative square root is represented by . The negative square root is represented by – .

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.

Reading Math

Example 1: Finding Square Roots of Perfect Squares

Find each square root.

42 = 16

32 = 9

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.

A.

= 4

B.

= –3

Find the square root.

Think: What number squared equals ?25

81

Positive square root positive .59

Example 1C: Finding Square Roots of Perfect Squares

Find the square root.

Check It Out! Example 1

22 = 4 Think: What number squaredequals 4?

Positive square root positive 2. = 2

52 = 25 Think: What is the opposite of the square root of 25?

1a.

1b.

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 willneed to make a square covered in glitter.Her tube of glitter covers 13 square inches. What is the greatest side lengthShonda’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.

Understand the problem11

22 Make a Plan

The side length of the square is because

13. Because 13 is not a perfectsquare, is not a whole number. Estimate

to the nearest tenth.

=

Find the two whole numbers that isbetween. Because 13 is between the perfectsquares 0 and 16. is between and , or between 3 and 4.

Example 2 Continued

3 4

Because 13 is closer to 16 than to 9, is closer to 4 than to 3.

You can use a guess-and-check method to estimate .

Example 2 Continued

Guess 3.6: 3.62 = 12.96 too low Guess 3.7: 3.72 = 13.69 too high

is greater than 3.6.

is less than 3.7.

Solve33Example 2 Continued

3.6 3.7 43

Because 13 is closer to 12.96 than to

13.69, is closer to 3.6 than to 3.7. 3.6

A square with a side length of 3.6 incheswould have an area of 12.96 square inches.Because 12.96 is close to 13, 3.6 inchesis a reasonable estimate.

Look Back44

Example 2 Continued

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?

Check It Out! Example 2

Guess 6.2 6.22 = 38.44 too high

Guess 6.1 6.12 = 37.21 too low

Use a guess and check method to estimate .

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 ,

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

, , .

ab

12

71

910

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

ab

Example 3: Classifying Real Numbers

Write all classifications that apply to each Real number.

A. –32

–32 = – = –32.0

32 1

32 can be written as a fraction and a decimal.

rational number, integer, terminating decimal

B. 5

5 = = 5.051

5 can be written as a fraction and a decimal.

rational number, integer, whole number, naturalnumber, terminating decimal

Write all classifications that apply to each real number.

3a. 7 49

rational number, repeating decimal

3b. –12

rational number, terminating decimal, integer

irrational number

Check It Out! Example 3

3c.

67 9 = 7.444… = 7.4

7 can be written as a repeating decimal.

49

–12 = – = –12.0 12 1

32 can be written as a fraction and a decimal.

= 3.16227766… The digits continue with no pattern.

Find each square root.

1. 2. 3. 4.12 -8 37

– 12

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 aninch. 8.2 in.

Lesson Quiz

Write all classifications that apply to each real number.

6. 1

7. –3.89

8.

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

rational, repeating decimal

irrational