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

2. Holt Algebra 1 1-5 Square Roots and Real Numbers Warm Up Simplify each expression. 1. 6 2 36 2. 11 2 121 3. (–9)( – 9) 814. 25 36 Write each fraction as a decimal. 5. 2525 5959 6. 7. 5 3838 8. –1 5656 0.4 5.375 0.5 –1.83

3. Holt Algebra 1 1-5 Square Roots and Real Numbers Evaluate expressions containing square roots. Classify numbers within the real number system. Objectives

4. Holt Algebra 1 1-5 Square Roots and Real Numbers square rootterminating decimal perfect square repeating decimal real numbersirrational numbers natural numbers whole numbers integers rational numbers Vocabulary

5. Holt Algebra 1 1-5 Square Roots and Real 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  4 = 4 2 = 16 = 4= 4 Positive square root of 16 (–4)( – 4) = (–4) 2 = 16 = –4 Negative square root of 16 –

6. Holt Algebra 1 1-5 Square Roots and Real Numbers 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 0202 1 1212 1004 2 9 3232 16 4242 25 5252 36 6262 49 7272 64 8282 81 9292 10 2 The nonnegative square root is represented by. The negative square root is represented by –.

7. Holt Algebra 1 1-5 Square Roots and Real Numbers 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

8. Holt Algebra 1 1-5 Square Roots and Real Numbers Example 1: Finding Square Roots of Perfect Squares Find each square root. 4 2 = 16 3 2 = 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

9. Holt Algebra 1 1-5 Square Roots and Real Numbers Find the square root. Think: What number squared equals ? 25 81 Positive square root positive. 5959 Example 1C: Finding Square Roots of Perfect Squares

10. Holt Algebra 1 1-5 Square Roots and Real Numbers Find the square root. Check It Out! Example 1 2 2 = 4 Think: What number squared equals 4? Positive square root positive 2. = 2 5 2 = 25Think: What is the opposite of the square root of 25? 1a. 1b. Negative square root negative 5.

11. Holt Algebra 1 1-5 Square Roots and 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.

12. Holt Algebra 1 1-5 Square Roots and Real Numbers 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. Example 2

13. Holt Algebra 1 1-5 Square Roots and Real Numbers 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

14. Holt Algebra 1 1-5 Square Roots and Real Numbers Guess 3.6: 3.6 2 = 12.96 too low Guess 3.7: 3.7 2 = 13.69 too high is greater than 3.6. is less than 3.7. Example 2 Continued 3.6 3.7 4 3 Because 13 is closer to 12.96 than to 13.69, is closer to 3.6 than to 3.7.  3.6

15. Holt Algebra 1 1-5 Square Roots and Real Numbers All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.

16. Holt Algebra 1 1-5 Square Roots and Real Numbers 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:,,. abab 1212 7171 9 10

17. Holt Algebra 1 1-5 Square Roots and Real Numbers 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:,,  abab

18. Holt Algebra 1 1-5 Square Roots and Real Numbers 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 = = 5.0 5151 5 can be written as a fraction and a decimal. rational number, integer, whole number, natural number, terminating decimal

19. Holt Algebra 1 1-5 Square Roots and Real Numbers Write all classifications that apply to each real number. 3a. 7 4949 rational number, repeating decimal 3b. irrational number Check It Out! Example 3 67  9 = 7.444 … = 7.4 7 can be written as a repeating decimal. 4949 = 3.16227766 … The digits continue with no pattern.

20. Holt Algebra 1 1-5 Square Roots and Real Numbers Find each square root. 1. 2. 3. 4. 12 -8 3737 – 1212 5. The area of a square piece of cloth is 68 in 2. How long is each side of the piece of cloth? Round your answer to the nearest tenth of an inch. 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