Simplify each expression.

1. 16 2. 19

3. 25 4. 20

Tell which two whole numbers each square root falls between.

5. 6.

7. 8.

5 and 6

3 and 4

2 and 3

9 and 10

256 361

625 400

30 5

12 99

Vocabulary, Rational, Irrational, Classifying

Evaluating Expressions Involving Square Roots

Evaluate the expression.

Evaluate the square root.

Add.= 25

Multiply.= 18 + 7

3 36 + 7

3 36 + 7 = 3(6) + 7

Try This!

Evaluate the expression.

Evaluate the square root.

Add.= 14

Multiply.= 10 + 4

2 25 + 4

2 25 + 4 = 2(5) + 4

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: , , .

ab1

271

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

Real NumbersRational Numbers (Q)

Integers (Z)

Whole Numbers (W)

Natural Numbers (N)

Irrational Numbers

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.

Estimating Square Roots of Numbers

Each square root is between two integers. Name the integers. Explain your answer.

55Think: What are perfect squares close to 55?

72 = 49 49 < 55

82 = 64 64 > 55

55 is between 7 and 8 because 55 is between 49 and 64.

90–

Estimating Square Roots of Numbers

Think: What are perfect squares close to 90?

–92 = 81 81 < 90

–102 = 100 100 > 90

Each square root is between two integers. Name the integers. Explain your answer.

is between –9 and –10 because 90 is between 81 and 100.

90–

TRY THIS!

Each square root is between two integers. Name the integers. Explain your answer.

80Think: What are perfect squares close to 80?

82 = 64 64 < 80

92 = 81 81 > 80

80 is between 8 and 9 because 80 is between 64 and 81.

45–

Each square root is between two integers. Name the integers.

Think: What are perfect squares close to 45?

–62 = 36 36 < 45

–72 = 49 49 > 45

Try This!

is between –6 and –7 because 45 is between 36 and 49.

45–

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 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 perfect

square, is not a whole number. Estimateto the nearest tenth.

=

Find the two whole numbers that is between. Because 13 is between the perfect squares 9 and 16. is between and , or between 3 and 4.

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 . .

Estimating Real Numbers

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.

Solve33

Estimating Real Numbers

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

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 to the nearest tenth?

Try This!

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.

Classifying Real Numbers

Write all classifications that apply to each Real number.A. –32

–32 = – = –32.0 32 can be written as a fraction and a decimal.

rational number, integer, terminating decimal

B. 5

5 = = 5.05 can be written as a fraction and a decimal.

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

51

321

C. 69 The digits continue with no pattern.

69 8306623863.irrational number

9

–35.9 is a terminating decimal.–35.9

rational number, terminating decimal

= = 39 3

81 3

A.

B.

C.

9 = 3

Classifying Real Numbers

Write all classifications that apply to each Real number.

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

813

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

Write all classifications that apply to each real number.

3a. 7 4

9

rational number, repeating decimal3b. –12

rational number, terminating decimal, integer

irrational number

Try This!

3c.

67 9 = 7.444… = 7.4

7 can be written as a repeating decimal.

49

–12 = – = –12.0 -12 can be written as a fraction and a decimal.

= 3.16227766… The digits continue with no pattern.

121

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