Warm Up Simplify each expression. 1. 2. 3. 4. 16 2. 19 3. 25 4. 20 Tell which two whole numbers each square root falls between. 5 and 6 6. 2 and 3 5. 3 and 4 7. 8. 9 and 10

Vocabulary, Rational, Irrational, Classifying Real Numbers Vocabulary, Rational, Irrational, Classifying

Evaluating Expressions Involving Square Roots Evaluate the expression. 3 36 + 7 3 36 + 7 = 3(6) + 7 Evaluate the square root. = 18 + 7 Multiply. = 25 Add.

Try This! Evaluate the expression. 2 25 + 4 2 25 + 4 = 2(5) + 4 Evaluate the square root. = 10 + 4 Multiply. = 14 Add.

Vocabulary 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: , , .

Vocabulary (continued) 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: , , 

Real Number System (Values on a number Line) Rational Numbers (Q) Integers (Z) Irrational Numbers Whole Numbers (W) Natural Numbers (N)

Estimating Real 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. Think: What are perfect squares close to 55? 55 72 = 49 49 < 55 82 = 64 64 > 55 55 is between 7 and 8 because 55 is between 49 and 64.

Estimating Square Roots of Numbers Each square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 90? – 90 –92 = 81 81 < 90 –102 = 100 100 > 90 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. Think: What are perfect squares close to 80? 80 82 = 64 64 < 80 92 = 81 81 > 80 80 is between 8 and 9 because 80 is between 64 and 81.

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? Understand the problem 1 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.

Continued 13. Make a Plan The side length of the square is because = 2 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 9 and 16. is between and , or between 3 and 4.

Estimating Real Numbers 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 . .

Estimating Real Numbers Solve 3 Guess 3.6: 3.62 = 12.96 too low is greater than 3.6. Guess 3.7: 3.72 = 13.69 too high is less than 3.7. 3 3.6 3.7 4 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 inches Continued Look Back 4 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.

Try This! 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? Use a guess and check method to estimate . Guess 6.1 6.12 = 37.21 too low is greater than 6.1. Guess 6.2 6.22 = 38.44 too high 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.

Real numbers 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 can be written as a fraction and a decimal. –32 = – = –32.0 rational number, integer, terminating decimal B. 5 5 can be written as a fraction and a decimal. 5 = = 5.0 rational number, integer, whole number, natural number, terminating decimal C. The digits continue with no pattern. irrational number

Classifying Real Numbers Write all classifications that apply to each Real number. A. 9 9 = 3 rational number, integer, whole number, natural number, terminating decimal B. –35.9 –35.9 is a terminating decimal. rational number, terminating decimal = = 3 9 3 81 3 C. rational number, integer, whole number, natural number, terminating decimal

Write all classifications that apply to each real number. Try This! Write all classifications that apply to each real number. 7 can be written as a repeating decimal. 49 3a. 7 4 9 67  9 = 7.444… = 7.4 rational number, repeating decimal 3b. –12 -12 can be written as a fraction and a decimal. –12 = – = –12.0 rational number, terminating decimal, integer 3c. The digits continue with no pattern. = 3.16227766… irrational number

5. The area of a square piece of cloth is 68 in2. Lesson Quiz Find each square root. 3 7 1 2 -8 1. 12 2. 3. 4. – 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 7. –3.89 8. irrational

Homework Worksheet