Square Roots and Real Numbers 1-5 Square Roots and Real Numbers Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz

Warm Up Simplify each expression. 1. 62 2. 112 121 36 25 36 3. (–9)(–9) 81 4. Write each fraction as a decimal. 2 5 5 9 5. 0.4 6. 0.5 5 3 8 –1 5 6 7. 5.375 8. –1.83

Objectives Evaluate expressions containing square roots. Classify numbers within the real number system.

Vocabulary square root terminating decimal perfect square repeating decimal real numbers irrational 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. = 4 Positive square root of 16 4  4 = 42 = 16 (–4)(–4) = (–4)2 = 16 – = –4 Negative square root of 16

The nonnegative square root is represented by 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. 1 4 9 16 25 36 49 64 81 100 02 12 22 32 42 52 62 72 82 92 102

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. A. Think: What number squared equals 16? 42 = 16 Positive square root positive 4. = 4 B. Think: What is the opposite of the square root of 9? 32 = 9 = –3 Negative square root negative 3.

Example 1C: Finding Square Roots of Perfect Squares Find the square root. Think: What number squared equals ? 25 81 Positive square root positive . 5 9

Check It Out! Example 1 Find the square root. 1a. 22 = 4 Think: What number squared equals 4? = 2 Positive square root positive 2. 1b. 52 = 25 Think: What is the opposite of the square root of 25? Negative square root negative 5.

The square roots of many numbers like , are not whole 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.

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? 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.

Example 2 Continued 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 0 and 16. is between and , or between 3 and 4.

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

Example 2 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.

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? 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.

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: , , . a b 1 2 7 9 10

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: , ,  a b

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

Write all classifications that apply to each real number. Check It Out! Example 3 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 32 can be written as a fraction and a decimal. –12 = – = –12.0 12 1 rational number, terminating decimal, integer 3c. The digits continue with no pattern. = 3.16227766… irrational number

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 7. –3.89 rational, repeating decimal 8. irrational