Remember It? 85 Rev. MA.7.A Exponents MA.8.A Integer Exponents

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1 Exponents and Roots FLORIDA CHAPTER 4 Name Class Date Chapter at a Glance Benchmark Lesson Worktext Student Textbook Remember It? 85 Rev. MA.7.A Exponents MA.8.A Integer Exponents MA.8.A Scientific Notation MA.8.A Laws of Exponents MA.8.A Squares and Square Roots MA.8.A Estimating Square Roots MA.8.A Operations with Square Roots MA.8.A The Real Numbers MA.8.G The Pythagorean Theorem MA.8.G Applying the Pythagorean Theorem and Its Converse Study It! Write About It! CHAPTER 4 Chapter 4 Exponents and Roots 83

2 Vocabulary Connections LA The student will relate new vocabulary to familiar words. Key Vocabulary Vocabulario Vokabilè hypotenuse hipotenusa ipoteniz irrational number número irracional nonm irasyonèl perfect square cuadrado perfecto kare pafè Pythagorean Theorem teorema de Pitágoras Teorèm Pythagò (Pythagore) real number número real nonm reyèl scientific notation notación cientifica notasyon syantifik To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. CHAPTER 4 1. The word irrational contains the prefix ir-, which means not. Knowing what you do about rational numbers, what do you think is true of irrational numbers? 2. The word real means actual or genuine. How do you think this applies to real numbers that we speak of in math? 84 Chapter 4 Exponents and Roots

3 4-1 Remember It? Review skills and prepare for future lessons. Lesson 4-1 Exponents (Student Textbook pp ) Rev of MA.7.A.3.2 Write in exponential form Simplify ( 5 3 ) - ( ) = -125 Identify how many times 4 is used as a factor. Find the product of three 5 s and then make the answer negative. Write in exponential form ( -3 ) ( -3 ) x x x 6. 2n 2n 2n Simplify ( -1 ) ( -2 ) (-2) ( 4 5) Evaluate k 2 + 3k for k = -2. Notes Lesson Tutorial thinkcentral.com Chapter 4 Exponents and Roots 85

4 Notes 86 Chapter 4 Exponents and Roots

5 4-2 Explore It! MA.8.A.6.1 Use exponents to write large and small numbers and vice versa and to solve problems. Integer Exponents Explore Patterns in Exponents A power is a product made of repeated factors. In a power, an exponent is used to tell how many times a base is used as a factor. REMEMBER 3 is the base. 2 is the exponent. 3 2 = 3 3 Activity 1 1 Complete the table and observe the patterns. Exponential Form Base Exponent Number of Factors Expanded Form Value x 4 x a 3 Try This Write the value of the power described base 2, number of factors 5 2. expanded form exponent 4, base 3 4. number of factors 4, base Integer Exponents 87

6 Draw Conclusions 5. Is 2 5 the same as 2 5? Explain. The exponents in Activity 1 were all positive whole numbers. You can use number patterns to discover how to use negative exponents. Activity 2 1 Complete the tables. Some of the exponents are negative. Exponential Form Expanded Form Value Exponential Form Expanded Form Value Try This Find the value of each power Draw Conclusions 10. Explain how to find the value of a number raised to the -5 power Integer Exponents

7 4-2 MA.8.A.6.1 Use exponents to write large and small numbers and vice versa and to solve problems. Integer Exponents (Student Textbook pp ) Lesson Objectives Simplify expressions with negative exponents and evaluate the zero exponent = = = Example 1 Simplify. Write in decimal form. -2 A = B = = 10-1 = 10-2 = 10-1 = Check It Out! Simplify. Write in decimal form. 1a b. 10 Lesson Tutorial thinkcentral.com 4-2 Integer Exponents 89

8 Example 2 A. Simplify = 1 Write the in the numerator and in the denominator. 5-3 = = Find the product. B. Simplify ( -10 ) -3 (-10) -3 = 1 Write the in the numerator and in the denominator. (-10) -3 = 1 (-10) -3 = Find the product. Check It Out! 2a. Simplify ( -3 ) -5. 2b. Simplify ( -4 ) Integer Exponents Lesson Tutorial thinkcentral.com

9 Example 3 Simplify 5 - ( 6-4 ) -3 + (-2) ( 6-4 ) -3 + (-2) 0 = 5 - ( 2 ) -3 + (-2) 0 Subtract inside the. = 5 - ( ) + 1 Evaluate the. = Add and subtract from left to right. Check It Out! 3a. Simplify 10 + ( ) b. Simplify ( 7-3 ) Lesson Tutorial thinkcentral.com 4-2 Integer Exponents 91

10 4-2 Explore It! Integer Exponents LA The student will organize information to show understanding or relationships Think and Discuss 1. Express 1_ 2 using a negative exponent. 2. Tell whether an integer raised to a negative exponent can ever be greater than 1. Justify your answer. 3. Get Organized Complete the graphic organizer. Fill in the boxes by writing an example of an expression with each type of exponent. Show how to simplify each expression that you write. Simplifying Powers Positive Exponent Negative Exponent Zero Exponent Integer Exponents

11 4-2 Integer Exponents MA.8.A.6.1 Use exponents...to write large and small numbers and vice versa and to solve problems. Simplify. Write in decimal form Simplify. 3. (-4) 3 4. (-6) (-8) (-7) ( -4 ) ( 5-1 ) ( 6-3 ) ( ) ( ) ( 1-3 ) 5 ( -4 ) Evaluate each expression for the given value of the variable. 23. x -3 for x = y -2 for y = w for w = ( s -2 ) for s = ( -3t ) 2 for t = b -2-6 for b = Write an expression for the product of eight and x, raised to the negative third power. Then evaluate the expression for x = Write an expression for the difference of four and x, raised to the negative second power. Then evaluate the expression for x = Integer Exponents 93

12 4-2 Explore It! Integer Exponents MA.8.A.6.1 Use exponents to write large and small numbers and vice versa and to solve problems. 1. The weight of one dust particle is 10-7 gram. Write this measure in standard notation. 2. The northern yellow bat is one of Florida s larger bat species. An adult has a wingspan of about 14 inches and weighs between 3 ( 2 ) -3 and 3 ( 2 ) -2 ounces. Simplify these expressions. Use the table for 6 8. Unit Size in meters centimeter (cm) 10-2 m millimeter (mm) 10-3 m micrometer (μm) 10-6 m nanometer (nm) 10-9 m Angstrom (Å) m 6. Human eyes can see a resolution of about 100 μm. Write this measure in meters. 3. Recall that the formula for the area of a circle is A = π r 2. How can you use negative exponents to solve this equation for π? 7. The size of a bacterium is about 50 nm. Write this measure in meters. 4. A ruby-throated hummingbird weighs about 3-2 ounce. Simplify A ruby-throated hummingbird breathes times per minute while at rest. Write the simplest form for this number of breaths per minute. 8. Gridded Response An Angstrom is equal to 100,000 femtometers. What exponent of base 10 is used to express the size of a femtometer in meters? Integer Exponents

13 4-3 Explore It! Scientific Notation MA.8.A.6.1 Use exponents and scientific notation to write large and small numbers and vice versa and to solve problems. Explore Products and Powers of Ten You will investigate how to use powers of 10 to help you write very large numbers. Activity 1 1 Complete the table by writing the value of each power of 10. Power of Value Complete the table by finding the indicated product or factor. Factors Factor Power of Ten Product , ,750,000 Try This Find the indicated power of = = 21, = = 7, Scientific Notation 95

14 You can also use powers of 10 to help you write very small numbers. Activity 2 1 Complete the table by writing the decimal value of each power of 10. Power of Value Complete the table by finding the indicated product or factor. Factors Factor Power of Ten Product Try This Find the indicated power of = = = = Draw Conclusions 9. Describe how to multiply by a power of 10 with a positive exponent. 10. Describe how to multiply by a power of 10 with a negative exponent Scientific Notation

15 4-3 MA.8.A.6.1 Use exponents and scientific notation to write large and small numbers and vice versa and to solve problems. Scientific Notation (Student Textbook pp ) Lesson Objectives Express large and small numbers in scientific notation and compare two numbers written in scientific notation Vocabulary scientific notation Example 1 Write each number in standard notation. A = Think: Move the decimal right places. B = by the reciprocal. Think: Move the decimal C places = Think: Move the decimal right places. Lesson Tutorial thinkcentral.com 4-3 Scientific Notation 97

16 Check It Out! Write each number in standard notation. 1a b c Example 2 Write in scientific notation Move the decimal to get a number between and Set up notation. Think: The decimal needs to move left to change 7.09 to , so the exponent will be. Think: The decimal needs to move places. So written in scientific notation is. Check = = Check It Out! 2. Write in scientific notation Scientific Notation Lesson Tutorial thinkcentral.com

17 Example 3 A pencil is 18.7 cm long. If you were to lay 10,000 pencils of this length end-to-end, how many millimeters long would they be? Write the answer in scientific notation. 1 cm = mm, so 18.7 cm = mm. 187 mm 10,000 Find the total length.. Think: The decimal needs to move places to the. In scientific notation, the pencils end-to-end would be mm long. Check It Out! 3. An oil rig can hoist 2,400,000 pounds with its main derrick. It distributes the weight evenly between 8 wire cables. What is the weight that each wire cable can hold? Write the answer in scientific notation. Example 4 One cell has a diameter of approximately meters. Another cell has a diameter of meters. Which cell has a greater diameter? ? Compare powers of Since powers of 10 are equal, compare the values between 1 and The has a greater diameter. Check It Out! 4. A certain cell has a diameter of approximately meters. A second cell has a diameter of meters. Which cell has a greater diameter? Lesson Tutorial thinkcentral.com 4-3 Scientific Notation 99

18 4-3 Explore It! Scientific Notation LA The student will organize information to show understanding or relationships Think and Discuss 1. Explain the benefits of writing numbers in scientific notation. 2. Describe how to write in standard notation. 3. Get Organized Complete the graphic organizer. Tell whether the exponent is positive, negative, or zero when each type of positive number is written in scientific notation. Writing Numbers in Scientific Notation Positive Number Exponent Example Greater than 0, less than 1 Greater than or equal to 1, less than 10 Greater than or equal to Scientific Notation

19 4-3 Scientific Notation MA.8.A.6.1 Use exponents and scientific notation to write large and small numbers and vice versa and to solve problems. Write each number in standard notation Write each number in scientific notation ,000, ,631, , ,005,000, Compare. Write >, <, or = The mass of Earth is approximately 5,980,000,000,000,000,000,000,000 kilograms. Write this number in scientific notation. 29. The mass of a specific dust particle is grams. Write this number in standard notation. 4-3 Scientific Notation 101

20 4-3 Explore It! Scientific Notation 1. In June 2001, the Intel Corporation announced that they could produce a silicon transistor that could switch on and off 1,500,000,000,000 times per second. Express the speed of the transistor in scientific notation. 2. With this transistor, computers will be able to do calculations in the time it takes to blink your eye. Express the number of calculations in standard notation. One light-year is approximately equal to 5,870,000,000,000 miles. Use this information and the table for Write your answers in scientific notation. Distance From Earth To Stars Star MA.8.A.6.1 Use exponents and scientific notation to write large and small numbers and vice versa and to solve problems. Constellation Distance (light-years) Sirius Canis Major 8 Canopus Carina 650 Alpha Centauri Centaurus 4 Vega Lyra How far in miles is Sirius from Earth? 3. The elements in this fast transistor are 20 nanometers long. A nanometer 1 is of a meter. Express the length 1,000,000,000 of an element in the transistor in meters in scientific notation. Use a dictionary to find the meanings of each numerical prefix. Then write the given measure in seconds using scientific notation. 4. micro; 8 microseconds 5. nano; 5 nanoseconds 9. How much farther is Canopus from Earth than Sirius? 10. How much closer is Alpha Centauri from Earth than Sirius? 11. Short Response Explain how to use scientific notation to express a light-year in miles. 6. pico; 6 picoseconds 7. femto; 2 femtoseconds Scientific Notation

21 4-4 Explore It! Laws of Exponents Multiply and Divide Powers You can use patterns in tables of numbers to discover rules for multiplying and dividing numbers written in exponential form. MA.8.A.6.3 Simplify real number expressions using the laws of exponents. Activity 1 Complete the table. Use the information in Column 2 to write the product as a power in Column 3. Product Factors of Product Exponential Form Sum of Exponents in Column (4 4 4) (4 4) = ( ) ( ) = (3 3) (3 3 3) (6 6 6) (6 6 6) Try This Write each product as a single power. Example: = Draw Conclusions 5. Compare: Look at the exponent in the third column and the sum in the fourth column. How are they alike? 6. Describe how you can multiply powers with the same base, like those in the first column. 4-4 Laws of Exponents 103

22 Activity 2 Complete the table. Be sure to eliminate common factors in Column 2. Quotient Factors of Product Quotient Written as a Power Difference of Exponents in Column = = Try This Write the quotient as a single power. Example: 37 3 = Draw Conclusions 11. Compare: Look at the exponent in the third column and the difference in the fourth column. How are they alike? 12. Describe how you can divide powers with the same base Laws of Exponents

23 4-4 MA.8.A.6.3 Simplify real number expressions using the laws of exponents. Laws of Exponents (Student Textbook pp ) Lesson Objectives Apply the laws of exponents Example 1 Multiply. l Write the product as one power. A B. n 5 n exponents. n exponents. C D exponents exponents. Check It Out! Multiply. Write the product as one power. 1a b. x 4 x 2 1c d. p 2 p 2 Lesson Tutorial thinkcentral.com 4-4 Laws of Exponents 105

24 Example 2 Divide. id Write the quotient as one power. A exponents. B. x 10 x 9 x 10-9 Subtract. Think: x 1 = Check It Out! Divide. Write the quotient as one power. 2a. 99 2b. n n 5 Example 3 Simplify. A. ( 5 4 ) 2 B. ( 6 7 ) exponents exponents. C. ( ( 2_ 3 ) 12 ) -3 D. ( 17 2 ) -20 ( 2_ 3) 12-3 exponents exponents Laws of Exponents Lesson Tutorial thinkcentral.com

25 Check It Out! Simplify. 3a. ( 7 2 ) -11 3b. ( 7-1 ) 2 3c. [( 1 5) 4 ] -2 3d. ( 5-2 ) -3 Example 4 The speed of sound at sea level is meters per second. A ship that is 5 kilometers offshore sounds its horn. About how many seconds will pass before a person standing on shore will hear the sound? Write your answer in scientific notation. distance = rate time 5 km = ( ) t = ( ) t Write 5 km as meters. = ( ) t Write in scientific notation = 2 t Divide both sides by. = t Write as a product of quotients t Simplify each quotient. It would take about seconds for the sound to reach the shore. Check It Out! 4. The diameter of a red blood cell is about millimeters. Thai has a slide with a 2-cm drop of red blood cells on it. Approximately how many cells are on the slide? Write your answer in scientific notation. (Hint: Find the ratio of the size of the drop, in millimeters, to the size of one cell.) Lesson Tutorial thinkcentral.com 4-4 Laws of Exponents 107

26 4-4 Explore It! Laws of Exponents LA The student will organize information to show understanding or relationships Think and Discuss 1. Explain why the exponents cannot be added in the product List two ways to express 4 5 as a product of powers. 3. Get Organized Complete the graphic organizer. Fill in the boxes by writing an example that illustrates each Law of Exponents. Laws of Exponents Multiplying Powers with the Same Base Dividing Powers with the Same Base Raising a Power to a Power Laws of Exponents

27 4-4 Laws of Exponents MA.8.A.6.3 Simplify real number expressions using...the laws of exponents. Multiply. Write the product as one power a 8 a (-w) 8 (-w) (-12) 18 (-12) w 14 w 12 Divide. Write the quotient as one power. 9. a 25 a 10. (-13)14 18 (-13) (-x ) 17 (-x ) 7 Write the product or quotient as one power r 9 r x x ( -17 ) 8 ( -17 ) m 16 m ( -b ) 21 ( -b ) Hampton has a baseball card collection of 5 6 cards. He organizes the cards into boxes that hold 5 4 cards each. How many boxes will Hampton need to hold the cards? Write the answer as one power. 24. Write the expression for a number used as a factor seventeen times being multiplied by the same number used as a factor fourteen times. Then write the product as one power. 25. After 3 hours, a bacteria colony has ( 25 2 ) 3 bacteria present. How many bacteria are in the colony? Write your answer in standard form. 4-4 Laws of Exponents 109

28 4-4 Explore It! MA.8.A.6.3 Simplify real number expressions using the laws of exponents. Laws of Exponents 1. A researcher separated her fruit flies into 2 2 jars. She estimates that there are 2 10 fruit flies in each jar. How many fruit flies does the researcher have in all? Use the table for 6 and 7. The table describes the number of people involved at each level of a pyramid pattern. In this pyramid pattern, each individual recruits 5 others to participate, who in turn recruit 5 others, and so on. 2. Suppose a researcher tests a new method of pasteurization on a strain of bacteria in his laboratory. If the bacteria are killed at a rate of 8 9 per second, how many bacteria would be killed after 8 2 second? Pyramid Pattern Level Total Number of People A satellite orbits the earth at about 13 4 km per hour. How long would it take to complete 24 orbits, which is a distance of about 13 5 km? (Hint: Use d = rt, distance equals rate times time.) 4. The side of a cube is 3 4 centimeters long. What is the volume of the cube? (Hint: V = s 3 ) 5. The wavelengths of electromagnetic radiation vary greatly. Green light has a wavelength of about meters. The wavelength of a U-band radio wave is meters. About how many times greater is the wavelength of a U-band radio wave than that of a green light? Justify your answer. 6. How many levels will it take to exceed 100,000 people? 7. How many times more people will be involved at level 6 than at level 2? 8. Short Response Belize borders Mexico and Guatemala in Central America. It has an area of square kilometers. Russia borders fourteen countries and is times larger than Belize. What is the area of Russia? Write your answer in scientific notation. Show your work Laws of Exponents

29 4-5 Explore It! Squares and Square Roots Relate Squares and Square Roots To square a number means to multiply the number by itself. MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems side = 5 5 squared = 5 5 = 5 2 = is the area of a square that is 5 units long on a side. In these activities, you will investigate the relationship between squares and square roots. area = 5 2 = 25 Activity 1 1 Use square tiles. Make a square that measures 6 tiles on a side. How many tiles did you use? 2 Make a square that measures 4 tiles on a side. How many tiles did you use? 3 On the grid at the right, draw squares 3 units on a side and 7 units on a side. Inside each square, write the area of the square. 4 Complete the table. Number Number Squared Expanded Form Evaluate Try This Evaluate each square squared Draw Conclusions 5. Explain why 4 2 equals 16 and does not equal Squares and Square Roots 111

30 When you find two equal factors of a number, you have found a square root of the number. 7 7 = 49, so 7 is a square root of 49. Since (-7) (-7) = 49, -7 is also a square root of 49. Use a radical symbol to indicate the nonnegative square root: 49 = 7. Activity 2 1 Use 16 square tiles to make a square. How many tiles are on each side of the square? 2 What happens if you try to make a square using 20 square tiles? 3 Complete the table. Number Number Squared Square Root of Number Use the symbol - number to indicate the negative square root of a number. 4 Complete the table. Number Number Number Try This Evaluate each square root Draw Conclusions 10. Explain why both 10 and -10 are square roots of Squares and Square Roots

31 4-5 MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems Squares and Square Roots (Student Textbook pp ) Lesson Objective Find square roots Vocabulary square root principal square root perfect square Example 1 Find the two square roots of each number. A = is a square root, since 7 7 = = is also a square root, since (-7) (-7) =. B = is a square root, since = = is also a square root, since (-10) (-10) =. C = is a square root, since = = is also a square root, since (-15) (-15) =. Lesson Tutorial thinkcentral.com 4-5 Squares and Square Roots 113

32 Check It Out! Find the two square roots of each number. 1a. 81 1b c d. 361 Example 2 A square window has an area of 169 square inches. How wide is the window? Find the square root of Use the = 169 So 169 = The window is Check It Out! to find the length of the window. square root; a negative length has no meaning. inches wide. 2. A square window has an area of 225 square inches. How wide is the window? Squares and Square Roots Lesson Tutorial thinkcentral.com

33 Example 3 Simplify each expression. A = 3( ) + 7 Evaluate the root. = + 7 Multiply. = Add. B = Rewrite as. = Evaluate the roots. = Add. Check It Out! Simplify each expression. 3a b c d Lesson Tutorial thinkcentral.com 4-5 Squares and Square Roots 115

34 4-5 Explore It! Squares and Square Roots LA The student will organize information to show understanding or relationships Think and Discuss 1. Describe what is meant by a perfect square. Give an example. 2. Explain how many square roots a positive number can have. How are these square roots different? 3. Decide how many square roots 0 has. Tell what you know about square roots of negative numbers. 4. Get Organized Complete the graphic organizer. Fill in the boxes by writing the principal square root and negative square root of 36. For each of these, write the square root using a radical symbol ( ) and as an integer. Principal Square Root Square Roots of 36 Negative Square Root Radical Integer Radical Integer Squares and Square Roots

35 4-5 Squares and Square Roots MA.8.A.6.2 Make reasonable approximations of square roots..., and use them to estimate solutions to problems... Find the two square roots for each number Evaluate each expression Find the product of six and the sum of the square roots of 100 and Explain how you can verify that 289 is 17 without using a calculator. 23. If a replica of the ancient pyramids was built with a base area of 1024 in 2, what would be the length of each side? (Hint: s = A ) 24. The maximum displacement speed of a boat is found using the formula: Max Speed (km/h) = 4.5 waterline length (m). Find the maximum displacement speed of a boat that has a waterline length of 9 meters. 4-5 Squares and Square Roots 117

36 4-5 Explore It! Squares and Square Roots MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems Use the table for 1 3. Amateur Wrestling Square Mat Sizes Division Size Home Use - small 100 ft 2 Home Use - large 144 ft 2 High School Competition 1444 ft 2 NCAA College Competition 1764 ft 2 1. What is the length of each side of the wrestling mat for NCAA College competition? A carpenter wants to use as many of her 82 small wood squares as possible to make a large square inlaid box lid. Use this information for 6 and How many squares can the carpenter use? How many squares would she have left? 7. How many more small wood squares would the carpenter need to make the next larger possible square box lid? 2. What is the length of each side of the wrestling mat for High School Competition? 3. Compare the mats for home use. How much smaller is one side of a small mat than one side of a large mat? 4. A middle school plans to purchase a 32-ft by 32-ft practice mat. If the estimated cost is $3.50 $4.50 per square foot, how much can they expect to pay for the new mat? 8. When the James family moved into a new house they had a square area rug that was 132 square feet. In their new house, there are three bedrooms. Bedroom one is 11 feet by 11 feet. Bedroom two is 10 feet by 12 feet, and bedroom three is 13 feet by 13 feet. In which bedroom will the rug fit? 9. Gridded Response A box of tiles contains 12 tiles. If you tile a square area using whole tiles, how many tiles will you have left from the box? 5. The Japanese art of origami requires folding square pieces of paper. Elena begins with a large sheet of square paper that is 169 in 2. How many 4-in. 4-in. squares can she cut out of the paper? Squares and Square Roots

37 Got It? Ready to Go On? Go to thinkcentral.com 4-2 THROUGH 4-5 Quiz for Lessons 4-2 through Integer Exponents (Student Textbook pp ) Simplify ( -3 ) ( 5 ) ( ) 4-3 Scientific Notation (Student Textbook pp ) Write each number in scientific notation ,980, Write each number in standard notation According to the US Census Bureau, the population of Florida in 2000 was nearly 16 million people, and the per capita income was approximately $21,500. Write the estimated total income in 2000 for Florida residents in scientific notation. 4-4 Laws of Exponents (Student Textbook pp ) Simplify. Write the product or quotient as one power q 9 q ( 3 3 ) ( 4 2 ) ( -x 2 ) ( 4-2 ) The mass of the known universe is about solar masses, which is metric tons. How many metric tons is one solar mass? 4-5 Squares and Square Roots (Student Textbook pp ) Find the two square roots of each number , If Jan s living room is 20 ft 16 ft, will a square rug with an area of 289 f t 2 fit? Justify your answer. Chapter 4 Exponents and Roots 119

38 4-2 THROUGH 4-5 Connect It! MA.8.A.6.1, MA.8.A.6.3, MA.8.A.6.4 Connect the concepts of Lessons 4-2 through 4-5 Match Up Work with a partner to play this matching game. 1. Write each of the following expressions on an index card. 2. Shuffle the cards. Lay them out face down on the desk in front of you. 3. You and your partner should take turns flipping over two cards. If the expressions on the cards are equal, the player who made the match keeps the cards and takes another turn. If the cards are not a match, the player turns the cards facedown and the other player takes a turn. 4. The game ends when there are no cards left. The winner is the player who collects the most cards. 5. Describe any strategies you used to help you play the game (2 2 ) 2 (3 2 ) Hop To It! Find a path that lands on each disk exactly once. You may start at any disk and then hop to any disk next to it. When you hop, you must always move to a greater number. 1. Draw your path along the disks. (2 2 ) (-2) Think About The Puzzler 2. Describe any strategies you used to find the path. 120 Chapter 4 Exponents and Roots

39 4-6 Explore It! Estimating Square Roots MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems Make Square Root Approximations A perfect square is a number that has integers as its square roots. 25 is a perfect square because 25 = 5 and - 25 = -5. The square roots, 5 and -5, are integers. You can use perfect squares to estimate square roots of numbers that are not perfect squares. 1 2 = = = = = = = = = = = = 36 Activity 1 1 Estimate the location of 11 on the number line. Think: 9 = 3 and 16 = 4. So, 11 must lie between 3 and Estimate the locations of 5, 20, 41, 68, and 90 on the number line. Write the square root along with an arrow pointing to its approximate location In Step 1, you estimated 11 as a number between 3 and 4. You can make a closer estimate by squaring numbers between 3 and 4. Use the table at the right. Between what two consecutive numbers, written to tenths, does 11 lie? and 4 Use your calculator to find each square = = = = = = = = = = = 6.52 = = = = = 5 Use your answers to Step 4 to estimate: Between what two consecutive numbers, written to tenths, does 45 lie? and 4-6 Estimating Square Roots 121

40 Try This Between what two consecutive whole numbers does the square root lie? Draw Conclusions 5. In Step 3 of Activity 1, you found two numbers, written to tenths, between which 11 lies. How could you find two numbers, written to hundredths, between which 11 lies? In Activity 2, you can use your calculator to estimate square roots with great accuracy. Activity 2 1 Complete the table using a calculator. Round square roots to the nearest thousandth. Number Square Root Solve using a calculator. Round square roots to the nearest thousandth. 2 the length of a side of a square with area 76 cm 2 3 the length of a side of a square photograph with area 39 in 2 4 the length of a side of a square postage stamp with area 172 mm 2 Try This Find the square root to the nearest thousandth Draw Conclusions 10. How could you find a number that has a square root that is an integer? Give an example Estimating Square Roots

41 4-6 MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems Estimating Square Roots (Student Textbook pp ) Lesson Objective Estimate square roots and solve problems using square roots Example 1 Each square root is between two consecutive integers. Name the integers. Explain your answer. A. 55 Think: Which squares are closest to 55? 7 2 = 49 < = 64 > is between and because is between and. B Think: Which perfect are closest to 90? (-9) 2 = 81 < 90 (-10) 2 = 100 > is between and because is between and. Check It Out! Each square root is between two consecutive integers. Name the integers. Explain your answer. 1a. 80 1b. 105 Lesson Tutorial thinkcentral.com 4-6 Estimating Square Roots 123

42 Example 2 You want to sew a fringe on a square tablecloth with an area of 500 square inches. Calculate the length of each side of the tablecloth and the length of fringe you will need to the nearest inch. List perfect squares near ,,, 576. Find the perfect squares nearest 500. < 500 < Find the square roots of the perfect squares. < 500 < < 500 < 500 is closer to than 529, so 500 is closer to. 500 Each side of the tablecloth is about inches. Now estimate the length around the tablecloth. 4 You will need about Check It Out! inches of fringe. 2. You want to build a fence around a square garden that is 250 square feet. Calculate the length of one side of the garden and the total length of the fence, to the nearest foot Estimating Square Roots Lesson Tutorial thinkcentral.com

43 Example 3 Estimate 141 to the nearest hundredth. Step 1: Find the value of the whole number. < 141 < Find the perfect squares nearest 141, and order the square roots of the perfect squares. < 141 < The number will be between and. The whole number part of the answer is. Step 2: Find the value of the decimal = Find the difference between the given number, 141, and the lower perfect square = Find the difference between the greater perfect square and the lower perfect square. Write the difference as a ratio, and find the approximate decimal value. Step 3: Find the approximate value. + = Combine the whole number and decimal. The approximate value of 141 to the nearest hundredth is. Check It Out! 3. Estimate 154 to the nearest hundredth. Example 4 Use a calculator l to find 600. Round to the nearest tenth. Using a calculator, Rounded, 600 is. Check It Out! 4. Use a calculator to find 800. Round to the nearest tenth. Lesson Tutorial thinkcentral.com 4-6 Estimating Square Roots 125

44 4-6 Explore It! Estimating Square Roots LA The student will organize information to show understanding or relationships Think and Discuss 1. Discuss whether 9.5 is a good first guess for Determine which square root or roots would have 7.5 as a good first guess. 3. Get Organized Complete the graphic organizer. Describe each method and give an example of each. Estimating Square Roots Between two integers To the nearest tenth With a calculator Estimating Square Roots

45 4-6 Estimating Square Roots MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems Each square root is between two consecutive integers. Name the integers. Explain your answer Estimate each square root to the nearest hundredth Simplify each expression ( 9 ) ( 15 ) Squaring and taking the square root are said to be inversely related. Explain what this means. 18. The area of a square tetherball court is 260 ft 2. What is the approximate length of each side of the court? Find your answer to the nearest foot. 19. Brian jogs one time around a square park with an area of 5 mi 2. About how far does Brian jog? 20. Steve wants to make a curtain to cover a square window with an area of 12 ft 2. About how long should each side of the curtain be? 4-6 Estimating Square Roots 127

46 4-6 Explore It! Estimating Square Roots MA.8.A.6.2 Make reasonable approximations of square roots, and use them to estimate solutions to problems... The distance d in kilometers to the horizon can be found using the formula d = h, where h is the height in kilometers above the ground. Use this information for 1 4. Estimate each distance and show your work. Then use a calculator to find an approximation to the nearest kilometer. 1. How far is it to the horizon when you are standing on the top of Mt. Everest, a height of 8.85 km? You can find the approximate speed of a vehicle that leaves skid marks before it stops. The table shows the minimum speed, in mi/h, that a vehicle was traveling before the brakes were applied. Use this table for 5 8. Length of Skid Marks L (ft) Minimum Speed S (mi/h) ( 10 ) ( 20 ) ( 30 ) Find the distance to the horizon from the top of Mt. McKinley, Alaska, a height of km. 3. How far is it to the horizon if you are standing on the ground and your eyes are 2 m above the ground? (Hint: 2 m = km) 4. Mauna Kea is an extinct volcano on Hawaii that is about 4 km tall. You should be able to see the top of Mauna Kea when you are how far away? Given the length of a vehicle s skid mark before stopping, find the minimum speed of the vehicle before it stopped. Round to the nearest mile per hour feet feet feet 8. Short Response The formula S = L, where S is the speed in miles per hour and L is the length of the skid marks in feet, gives the maximum speeds that the vehicle was traveling before the brakes were applied. Find the approximate range of speed that a vehicle leaving a 200-ft skid could have been traveling before the brakes were applied Estimating Square Roots

47 4-7 Explore It! MA.8.A.6.4 Perform operations on real numbers (including radicals and irrational numbers) using multistep and real world problems. Operations with Square Roots Explore Square Root Relationships In the following activities, you will explore operations with square roots and look for relationships between the numbers involved. Activity 1 1 How does the grid show that 4 9 = 36? 2 Now, take the square root of each part of the equation in Step 1. Is it true that 4 9 = 36? 3 To test whether the relationship holds true in other cases, complete the table. Do not use a calculator. Try This Simplify. m n m n m n = 6 36 = Operations with Square Roots 129

48 Draw Conclusions 5. Describe two ways you can find the product 9 4. Activity 2 1 Complete the first 2 rows by hand. Then use a calculator to complete the other rows. Round square roots to the nearest thousandth. m n m n m n = = = 4 = Try This Simplify Draw Conclusions 10. Describe two ways you can find the quotient Operations with Square Roots

49 4-7 MA.8.A.6.4 Perform operations on real numbers (including radicals and irrational numbers) using multistep and real world problems. Operations with Square Roots (Student Textbook pp ) Lesson Objective Use the laws of exponents to simplify square roots. Vocabulary radical expression radical symbol radicand Example 1 Simplify. A The are the same. = ( + ) 6 Combine like terms. = 6 B = = ( - ) 7 = Check It Out! Simplify. Combine like terms. Property 1a b Lesson Tutorial thinkcentral.com 4-7 Operations with Square Roots 131

50 Example 2 Simplify. A = = = B = 4 = 4 = = 80 Multiply the radicands under one symbol. Simplify. the radicands under one radical symbol. Simplify. Check It Out! Simplify. 2a b Example 3 Simplify 162 Method A Method B 162 = = 9 = = = = 2 = = Operations with Square Roots Lesson Tutorial thinkcentral.com

51 Check It Out! 3. Simplify 180. Example 4 Simplify. A = = If the radicand has any perfect squares, factor them out. Simplify. = 5 B = = + 3 = + 3 = Check It Out! Simplify. Factor any squares out of the radicands. Simplify. 4a b Lesson Tutorial thinkcentral.com 4-7 Operations with Square Roots 133

52 4-7 Explore It! Operations with Square Roots LA The student will organize information to show understanding or relationships Think and Discuss 1. Explain why you can combine terms in the expression , but you cannot combine terms in the expression Show two ways to factor 200 so that each way contains a different perfect square factor. 3. Get Organized Complete the graphic organizer. Fill in the boxes by giving an example of a square root that can be simplified and show how to simplify it. Then give an example of a square root that cannot be simplified. A Square Root That Can Be Simplified Simplifying Square Roots A Square Root That Cannot Be Simplified Simplified Version Operations with Square Roots

53 4-7 MA.8.A.6.4 Perform operations on real numbers (including...radicals...and irrational numbers) using multi-step and real world problems. Operations with Square Roots Simplify Find the area and perimeter of the rectangle. Write each answer in simplest form. Show your work. 75 in. 12 in. 4-7 Operations with Square Roots 135

54 4-7 Explore It! Operations with Square Roots MA.8.A.6.4 Perform operations on real numbers (including radicals and irrational numbers) using multi-step and real world problems. 1. Palm Beach County is approximately shaped like a square with area 2580 mi Anne wants to have a square garden with an area of 200 square feet. Write the length of each side of the square in simplest radical form. Inland Florida PALM BEACH BROWARD Atlantic Ocean Broward County borders Palm Beach County as shown. Suppose the State of Florida decided to change the borders of Broward County to be a square of area 645 mi 2. Find the length of the combined coastline that the two counties share. Write your answer in simplest radical form. 2. One student used 5 7 = 35 to approximate 1250 to the nearest whole number. Explain how the student arrived at A kicker for the Florida Gators kicked a football to a height of 128 ft into the air. Use the formula, t = h 4, where t is the time in seconds it takes for an object to fall from a height of h feet. About how long will it take for the football to come back down to the field? Write your answer in simplest radical form. 5. Gridded Response The landscaper for a park needs to know the area of the park so that she can buy enough materials. The plots for the Library and City Hall are both squares. What is the area of the park in square meters? (Hint: Find the dimensions of the City Hall and Library first.) Green Park City Hall 1200 m 2 Library 300 m Operations with Square Roots

55 4-8 Explore It! The Real Numbers MA.8.A.6.2 compare mathematical expressions involving real numbers and radical expressions. Explore Real Numbers Fractions, decimals, whole numbers, natural numbers, and integers are all rational numbers. In the following activities, you will explore real numbers that are not rational numbers. REMEMBER A rational number can be written as a ratio, a, where a and b are integers and b 0. b Activity 1 Complete the table. Show that each number is a rational number by writing it as a ratio of two integers a b. Number Ratio a b = = 7 = = = Try This 1. Show that 16 is a rational number. 2. Show that -8 3_ is a rational number Show that 6.25 is a rational number. 4-8 The Real Numbers 137

56 Activity 2 1 Locate each number on the number line. Draw a point there and write the letter of the number (A, B, C, etc.) above the point. (A) 0 5 (F) - 11 (B) 3 (C) (D) 7 3 (G) - 7 (H) (I) (E) - 2 (J) An irrational number is a number that can NOT be written as a ratio a of two b integers a and b. Square roots of whole numbers that are not perfect squares are irrational numbers. 2 Which numbers in Step 1 above are irrational numbers? Try This Graph each irrational number on the number line Draw Conclusions 7. Explain why every integer is a rational number. 8. Explain whether 3.5 is a rational number Give three examples of irrational numbers The Real Numbers

57 4-8 MA.8.A.6.2 compare mathematical expressions involving real numbers and radical expressions The Real Numbers (Student Edition pp ) Lesson Objectives Determine if a number is rational or irrational Vocabulary irrational number real number Density Property Example 1 Write all names that apply to each number. A. 5 5 is a number that is not a perfect. B is a decimal. C = 4 2 = 2 Lesson Tutorial thinkcentral.com 4-8 The Real Numbers 139

58 Check It Out! 1a. 9 Write all names that apply to each number. 1b c Example 2 State t if the number is rational, irrational, or not a real number. Justify your answer. A =, because is a whole number 3 B ; because it is the of a negative number C. 4 9 ( 2 3) ( 2 3) = 4 9, 2 is rational 3 Check It Out! 2a. -7 State if the number is rational, irrational, or not a real number. Justify your answer The Real Numbers Lesson Tutorial thinkcentral.com

59 2b c Example 3 Find a real number between and There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. ( ) 2 = 5 = 2 2 = Check It Out! A real number between 3 2_ 5 and 3 3 _ 5 is. 3. Find a real number between 5 1 and Lesson Tutorial thinkcentral.com 4-8 The Real Numbers 141

60 4-8 Explore It! The Real Numbers LA The student will organize information to show understanding or relationships Think and Discuss 1. Explain how rational numbers are related to integers. 2. Tell if a number can be irrational and whole. Explain. 3. Use the Density Property to explain why there are infinitely many real numbers between 0 and Get Organized Complete the graphic organizer. Fill in the flowchart by writing each of the following numbers in every box for which the classification applies: -2, 3, 0, 19, 2_ Irrational Rational No Yes 3. Integer Yes Whole number Yes Natural number No No No The Real Numbers

61 4-8 The Real Numbers MA.8.A.6.2 Compare mathematical expressions involving real numbers and radical expressions. Write all names that apply to each number State if the number is rational, irrational, or not a real number Find a real number between each pair of numbers and and Give an example of a rational number between - 36 and Give an example of an irrational number less than and Give an example of a number that is not real. 4-8 The Real Numbers 143

62 4-8 Explore It! The Real Numbers MA.8.A.6.2 Compare mathematical expressions involving real numbers and radical expressions. 1. Twin primes are prime numbers that differ by 2. Find an irrational number between twin primes 5 and 7. Use the number line for 6 8. For each point on the number line, write a possible rational and a possible irrational number that it could represent. 2. Rounded to the nearest ten-thousandth, π = Find a rational number between 3 and π. 6. A A B C One famous irrational number is e. Rounded to the nearest ten-thousandth e Find a rational number that is between 2 and e. 7. B 8. C 4. Perfect numbers are numbers in which the factors of the number (excluding the number itself) add up to the number itself. For example, the number 6 is a perfect number because = 6. The number 28 is also a perfect number. Find an irrational number between 6 and Write all the names that apply to any number that gives the average amount of rainfall for a week. 9. Short Response Explain when the length of a side of a square would be a rational number and when it would be an irrational number The Real Numbers

63 4-9 Explore It! MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. The Pythagorean Theorem Explore Right Triangles In Activity 1, you will explore an interesting relationship between the side lengths of a right triangle. REMEMBER A right triangle has one right angle. An isosceles triangle has two congruent sides. Activity 1 1 The drawing at the right shows an isosceles right triangle and three squares. Use grid paper to make a drawing like the one shown. You will be cutting out the pieces, so be sure your drawing is large enough that you can easily cut out and work with the pieces. 2 Cut out the two smaller squares, then cut those squares in half along a diagonal. Fit the pieces of the smaller squares on top of the large square. Try This 1. Compare your results with those of a classmate to confirm this relationship among the shapes. Draw Conclusions 2. Describe the relationship between the areas of the small squares and the large square. 3. How do the side lengths of the triangle relate to the areas of the squares? 4-9 The Pythagorean Theorem 145

64 Activity 2 1 Draw a right triangle with legs of 3 units and 4 units on graph paper. For each leg, draw a square that has a leg as one side. What is the sum of the areas of these two squares? 3? 4 2 Measure the length of the hypotenuse using graph paper. Draw a square with the hypotenuse length as one side of the square. What is its area? Compare the sum of the areas you found in Step 1 to the area you found in Step 2. How are they related? Try This Repeat Activity 2 for right triangles with legs of the given lengths. 4. 5, , , 15 Draw Conclusions 7. How are the square areas next to the sides of the triangle related to the sides of the triangle? 8. Describe the relationship between the areas of the small squares and the area of the large square The Pythagorean Theorem

65 4-9 The Pythagorean Theorem (Student Textbook pp ) Lesson Objective Use the Pythagorean Theorem to solve problems Vocabulary Pythagorean Theorem MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. leg hypotenuse Example 1 Find the length of each hypotenuse to the nearest hundredth. A. c a 2 + b 2 = c = c 2 Substitute for a and b. + = c 2 Simplify powers. Theorem = c 2 = c Solve for c ; c = c 2. c Lesson Tutorial thinkcentral.com 4-9 The Pythagorean Theorem 147

66 Find the length of each hypotenuse to the nearest hundredth. B. triangle with coordinates (1, -2), (1, 7), (13, -2) 20 y The points form a right triangle with a = 9 and b = (1, 7) 2 a 2 + b 2 = c 2 2 Use the Pythagorean Theorem + = c 2 Substitute 9 for a and 12 for b x -4O (1, -2) (13, -2) + = c 2 Simplify = c 2 = c Find the square root. Check It Out! Find the length of each hypotenuse to the nearest hundredth. 1a. 1b. triangle with coordinates c 5 ( 5, 4), (4, 4), and (4, 6) 7 Example 2 Solve for the unknown side in the right triangle to the nearest tenth. 25 b 7 - a 2 + b 2 = c b 2 = Theorem Substitute for a and c. + b 2 = Simplify powers. - b 2 = b = 576 = The Pythagorean Theorem Lesson Tutorial thinkcentral.com

67 Check It Out! 2. Solve for the unknown side in the right triangle. 12 b 4 Example 3 Two airplanes leave the same airport at the same time. The first plane flies to a landing strip 350 miles south, while the other plane flies to an airport 725 miles west. How far apart are the two planes after they land? 2 a 2 + b 2 = c 2 2 Pythagorean Theorem + = c 2 Substitute for a and b. + = c 2 Simplify. = c 2 The planes are about 805 miles apart after they land. Check It Out! c Find the square root. 3. Two birds leave the same spot at the same time. The first bird flies to his nest 11 miles south, while the other bird flies to his nest 7 miles west. How far apart are the two birds after they reach their nests? Lesson Tutorial thinkcentral.com 4-9 The Pythagorean Theorem 149

68 4-9 Explore It! The Pythagorean Theorem LA The student will organize information to show understanding or relationships Think and Discuss 1. Tell which side of a right triangle is always the longest side. 2. Explain if 2, 3, and 4 cm could be side lengths of a right triangle. 3. Get Organized Complete the graphic organizer. Fill in the boxes by writing the lengths of the legs and the length of the hypotenuse for the given right triangle. Then use these lengths to write an equation based on the Pythagorean Theorem. Right Triangle Lengths of Legs Length of Hypotenuse Pythagorean Theorem The Pythagorean Theorem

69 4-9 The Pythagorean Theorem MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. Find the length of each hypotenuse. Round to the nearest hundredth if necessary Solve for the unknown side in each right triangle. Round to the nearest tenth if necessary A rectangular swimming pool in a park is 60 feet long and 25 feet wide. Marsha swims from one corner of the pool to the opposite corner and back 10 times. How many feet does she swim? To meet federal guidelines, a wheelchair ramp that is constructed to rise 1 foot off the ground must extend 12 feet along the ground. How long will the ramp be? Round your answer to the nearest hundredth. 4-9 The Pythagorean Theorem 151

70 4-9 Explore It! The Pythagorean Theorem MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. 1. A 10-m tall utility pole is supported by two guy wires. Each guy wire reaches from the top of the poll down to the ground 3 meters away from the base of the pole. How many meters of wire are needed for the two guy wires? Round your answer to the nearest tenth. 5. A football field is 100 yards with 10 yards at each end for the end zones. The field is 45 yards wide. Use this information for 5 and 6. Round your answer to the nearest tenth. a. Find the length of the diagonal of the entire field, including the end zones. 2. A 12-foot ladder is resting against a wall. The base of the ladder is 2.5 feet from the base of the wall. How high up the wall will the ladder reach? Round your answer to the nearest tenth. 3. The glass for a picture window is 8 feet by 10 feet. The door it must pass through is 3 feet by 7.5 feet. Will the glass fit through the door? Justify your answer. 4. A television screen measures approximately 15.5 in. high and 19.5 in. wide. The television is advertised by giving the approximate length of the diagonal of its screen, to the nearest whole inch. What is the advertised size of this television? Show your work. b. How long would it take a player running at 22 ft per second to run the length of the diagonal? 6. The base-path of a baseball diamond forms a square. If it is 90 ft from home to first, how far does the catcher have to throw to catch someone stealing second base? Round your answer to the nearest tenth. 7. Short Response Hennrick is making a kite. The lengths of the rods for the frame are shown in the diagram. Find the perimeter of the kite to the nearest tenth. Show your work. 18 in. 27 in The Pythagorean Theorem

71 4-10 Explore It! Applying the Pythagorean Theorem and Its Converse MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. Explore the Pythagorean Theorem The Pythagorean Theorem states that for any right triangle with legs a and b and hypotenuse c, a 2 + b 2 = c 2. a c In the following activity, you will use the Pythagorean Theorem to find different right-triangle side lengths that are all integers. b a 2 + b 2 = c 2 Activity 1 Complete the table below. Then try choosing your own values for m and n for the last two rows. Use these rules for choosing m and n: 1. m and n should be positive whole numbers, with m > n. 2. One number should be odd and the other should be even. 3. m and n should not have any common factors. m n a = m 2 - n 2 b = 2mn c = m 2 + n 2 Does a 2 + b 2 = c 2? yes Applying the Pythagorean Theorem and Its Converse 153

72 2 The figure at the right shows that if the values of a, b, and c in the first row of the table (3, 4, and 5) are used to construct the three sides of a triangle, it will be a right triangle. Choose another row of the table. Use the values of a, b, and c in that row to construct the three sides of a triangle below Describe your results. Try This Use the given values of m and n to calculate a, b, and c. 1. m = 6, n = 5 2. m = 7, n = 4 3. m = 7, n = 2 a = a = a = b = b = b = c = c = c = Draw Conclusions 4. Three numbers a, b, and c have the property that a 2 + b 2 = c 2. Describe a triangle that has sides of length a units, b units, and c units Applying the Pythagorean Theorem and Its Converse

73 4-10 MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. Applying the Pythagorean Theorem and Its Converse (Student Textbook pp ) Lesson Objective Use the Distance Formula and the Pythagorean Theorem and its converse to solve problems Example 1 What is the diagonal length of the rectangular projector screen below? 7 ft 3 ft + = c 2 Use the. + = c 2 Simplify. = c 2 Add. = c Take the of both sides. c Find the. The diagonal length is about Check It Out! feet. 1. A square garden has a side length of 10 meters. What is the length of the diagonal of the garden, to the nearest hundredth? 4-10 Applying the Pythagorean Theorem and Its Converse 155

74 Example 2 Find the distances between the points to the nearest tenth. y 2 J -4-2 L O 2 4 x -2 K M -4 A. J and K J(-4, 0) and K(0, -3) Let J be ( x 2, y 2 ) and K be ( x 1, y 1 ). d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 Use the. = ( - ) 2 + ( - ) 2 Substitute. = ( ) Subtract. = + Simplify powers. = = Add, then take the square root. The distance between J and K is B. L and M L(4,0) and M(5, -3) Let L be ( x 2, y 2 ) and M be ( x 1, y 1 ). units. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 Use the. = ( - ) 2 + ( - ) 2 Substitute. = Subtract. = + Simplify powers. = Add, then take the square root. The distance between L and M is about units Applying the Pythagorean Theorem and Its Converse

75 Check It Out! Find the distance between the points to the nearest tenth. 2a. K and L 2b. J and M Example 3 Tell whether the given side lengths form a right triangle. A. 9, 12, 15 a 2 + b 2 = c 2 Compare a 2 + b 2 to c Substitute. + Simplify. = Add. The side lengths a right triangle. B. 8, 10, 13 a 2 + b 2 = c 2 Compare a 2 + b 2 to c Substitute. + Simplify. The side lengths Check It Out! Add. form a right triangle. Tell whether the given side lengths form a right triangle. 3a. 8, 11, 13 3b. 18, 24, Applying the Pythagorean Theorem and Its Converse 157

76 4-10 Explore It! Applying the Pythagorean Theorem and Its Converse LA The student will organize information to show understanding or relationships Think and Discuss 1. Make a conjecture about whether doubling the side lengths of a right triangle makes another right triangle. 2. Get Organized Complete the graphic organizer. Fill in the boxes by writing the statement of the Distance Formula. Then give an example of how to use the formula by giving coordinates of point A and point B and showing how to find the distance between the points. Statement Distance Formula Point A Point B Example Distance Between A and B Applying the Pythagorean Theorem and Its Converse

77 4-10 Applying the Pythagorean Theorem and Its Converse MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. Determine whether each triangle is a right triangle mm 20 mm 10 ft 6 ft 16 mm 7 ft m 30 m 25 m 17 in. 15 in. 8 in. Tell whether the given side lengths form a right triangle. 8. 7, 10, , 20, , 11, , 7, A basketball court is 94 feet long and 50 feet wide. What is the length of a diagonal of the basketball court, to the nearest tenth? Find the distance between each pair of points. 13. (3, 6) and (1, 2) 14. (-5, -2) and (-8, 3) 15. (-5, 3) and (5, 5) 16. (3, 4) and (-4, -1) 4-10 Applying the Pythagorean Theorem and Its Converse 159

78 4-10 Explore It! Applying the Pythagorean Theorem and Its Converse MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane. 1. Federal guidelines require that a wheelchair ramp must extend at least 12 units along the ground for every 1 unit off the ground that it rises vertically. What is the minimum slanted length of a wheelchair ramp that reaches the top of a 5-ft-high staircase? Explain. 4. Linda made triangular flags for the spirit club to wave. Each flag was a right triangle. One side was 1.5 feet long and another side was 2.2 feet long. She used fringed trim along the longest side of the each flag. What was the length of fringed trim that she sewed to each flag? Round to the nearest tenth of a foot. On a map, each unit on the grid represents a mile. Use the information for 2 and One city is located at (4, 8) and another city is located at (6, 12) on the grid. How many miles apart are the two cities? Round to the nearest tenth of a mile. Show your work. 3. A post office is located at (2, 2). Find a point that is 13 miles from the location of the post office. Show your work. 5. Jorge wants to build a support in the shape of a right triangle. He has one 9-foot board and one 4-foot board. What are the two possible lengths he needs for the third board? Round your answer to the nearest hundredth. 6. Extended Response Tony needs to use a ladder to get onto the roof of an 11-ft house. His ladder is 14 ft long. According to safety regulations, the base of the ladder should be placed 6 ft from the base of the house, and the ladder should extend at least 1 ft over the roofline. Can Tony safely use his ladder to climb onto the roof? Justify your answer Applying the Pythagorean Theorem and Its Converse

79 Got It? Ready to Go On? Go to thinkcentral.com 4-6 THROUGH 4-10 Quiz for Lessons 4-6 through Estimating Square Roots (Student Textbook pp ) Each square root is between two consecutive integers. Name the integers The area of a chess board is 110 square inches. Find the length of one side of the board to the nearest hundredth. 4-7 Operations with Square Roots (Student Textbook pp ) Simplify The Real Numbers (Student Textbook pp ) Write all names that apply to each number Give an example of an irrational number that is less than The Pythagorean Theorem (Student Textbook pp ) Find the missing length for each right triangle. Round your answer to the nearest tenth, if necessary. 16. a = 3, b = 6, c = 17. a =, b = 24, c = Applying the Pythagorean Theorem and Its Converse (Student Textbook pp ) Find the distance between the points. Round to the nearest tenth, if necessary. 18. (3, 2) and (11, 8) 19. (-1, -1) and (-3, 6) Tell whether the given side lengths form a right triangle , 9, , 14, 17 Chapter 4 Exponents and Roots 161

80 4-6 THROUGH 4-10 Connect It! MA.8.A.6.2, MA.8.A.6.3, MA.8.A.6.4, MA.8.G.2.4 Connect the concepts of Lessons 4-6 through 4-10 Spiraling Out of Control 1. On a sheet of graph paper, draw an isosceles right triangle (Triangle A) with legs 1 unit long. Find the length of the hypotenuse and record it in the table. 2. Draw a new isosceles right triangle (Triangle B) so that one of its legs is the hypotenuse of the first triangle, as shown. Find and record the length of the legs and hypotenuse of Triangle B. 3. Continue to draw new isosceles right triangles in this way and record the length of the legs and hypotenuse for each triangle. Simplify any expressions with square roots. 4. When you have drawn 8 triangles (Triangles A through H), look for patterns in your completed table. Without drawing any new triangles, predict the lengths of the legs of Triangles I, J, K, and L. C A 1 1 B A B A Triangle A B C D E F G H Legs 1 Hypotenuse Serving Leftovers 1. Find pairs of expressions in the figure that have the same value. When you find a matching pair, cross out the expressions. 2. When you have crossed out all the matching pairs, the leftover expression will tell you the number of pounds of turkey the average American eats each year. Estimate this value to the nearest tenth Think About The Puzzler 3. Explain how you found one of the matching pairs. 162 Chapter 4 Exponents and Roots

81 FLORIDA Name Class Study It! Vocabulary CHAPTER Date 4 Multi-Language Glossary Go to thinkcentral.com (Student Textbook page references) Density Property (176) principal square root.... (164) radicand (172) hypotenuse (180) Pythagorean Theorem... (180) real number (176) irrational number (176) radical expression (172) scientific notation (152) leg (180) radical symbol (172) square root (164) perfect square (164) Complete the sentences below with vocabulary words from the list above. 1. A(n) is a number that cannot be written as a fraction. 2. is a short-hand way of writing extremely large or extremely small numbers. 3. The states that the sum of the squares of the of a right triangle is equal to the square of the. Lesson 4-2 IInteger Exponents (Student Textbook pp ) Simplify. ( -3 )-2 MA.8.A = 1 1 = 1 (-3)-2 = 2 9 ( -3 ) Simplify ( -4 ) ( 9-7 ) ( 6-9 ) ( 5-7 )-2 Lesson Scientific Notation (Student Textbook pp ) S Write in scientific notation. Write in standard notation ,000 35,800 MA.8.A , Lesson Tutorial thinkcentral.com ,000,000 62, , Chapter 4 Exponents and Roots 163