Exponents 4-1. Lesson Objectives. Vocabulary. Additional Examples. Evaluate expressions with exponents. exponential form (p. 162) exponent (p.
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1 LESSON 4-1 Exponents Lesson Objectives Evaluate expressions with exponents Vocabulary exponential form (p. 16) exponent (p. 16) base (p. 16) power (p. 16) Additional Examples Example 1 Write in exponential form. A Identify how many times is a factor. B. ( 6) ( 6) ( 6) ( 6) ( 6) ( 6) Identify how many times ( 6) is used as a. Example Simplify. A Find the product of 3 s. 69 Holt McDougal Mathematics
2 LESSON 4-1 CONTINUED Simplify. B. ( 3) 5 ( 3) 5 Find the product of 3 s. Example 3 Evaluate x(y x z y ) x y for x 4, y, and z 3. x(y x z y ) x y 4( 4 3 ) 4 Substitute for x, for y, and for z. 4( ) Evaluate the. 4( ) 16 inside the parentheses. 16 from left to right. from left to right. Example 4 Use the expression 1 (n 3n) to find the number of diagonals in a 7-sided figure. 1 (n 3n) 1 (7 3 7) Substitute the number of sides for n. 1 ( ) Simplify inside the. 1 ( ) inside the parentheses.. 70 Holt McDougal Mathematics
3 LESSON 4-1 CONTINUED Check It Out! 1. Write in exponential form. 7 7 b b. Simplify. (5 ) 3. Evaluate z 7( x x y ) for x 5, y, and z Use the expression 1 (n 3n) to find the number of diagonals in a 4-sided figure. 71 Holt McDougal Mathematics
4 LESSON 4- Integer Exponents Lesson Objectives Simplify expressions with negative exponents and evaluate the zero exponent Additional Examples Example 1 Simplify. Write in decimal form. A B Example A. Simplify Write the ; change the of the exponent Find the product of 1 5 s. B. Simplify ( 10) Write the ; change the of the exponent Find the product of three s Simplify. 7 Holt McDougal Mathematics
5 LESSON 4- CONTINUED Example 3 Simplify 5 (6 4) 3 ( ) 0. 5 (6 4) 3 ( ) 0 5 ( ) 3 ( ) 0 Subtract inside the. 5 ( ) 1 Evaluate the. Add and subtract from left to right. Check It Out! 1. Simplify. Write in decimal form Simplify ( 4) Simplify 10 (5 3) Holt McDougal Mathematics
6 LESSON 4-3 Properties of Exponents Lesson Objectives Apply the properties of exponents Additional Examples Example 1 Multiply. Write the product as one power. A B. n 5 n 7 exponents. exponents. C. 5 D exponents. exponents. Example Divide. Write the quotient as one power. 5 A exponents. 10 B. x x 9 Subtract. Think: x 1 74 Holt McDougal Mathematics
7 LESSON 4-3 CONTINUED Example 3 Simplify. A. (5 4 ) B. (6 7 ) 9 exponents. exponents. C D. (17 ) 0 exponents. exponents. Check It Out! 1. Multiply. Write the product as one power Divide. Write the quotient as one power Simplify. (5 ) 3 75 Holt McDougal Mathematics
8 LESSON 4-4 Scientific Notation Lesson Objectives Express large and small numbers in scientific notation and compare two numbers written in scientific notation Vocabulary scientific notation (p. 174) Additional Examples Example 1 Write each number in standard notation. A B Think: Move the decimal right places by the reciprocal. C Think: Move the decimal 3 places Think: Move the decimal right places. 76 Holt McDougal Mathematics
9 LESSON 4-4 CONTINUED Example 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 Example 3 A pencil is 18.7 cm long. If you were to lay 10,000 pencils end-to-end, how many millimeters long would they be? Write the answer in scientific notation. 1 cm mm 18.7 cm mm Multiply each side by. 187 mm 10,000 Find the total length.. Set up scientific notation. Think: The decimal needs to move to change 1.87 to 1,870,000, so the exponent will be. Think: The decimal needs to move places. 77 Holt McDougal Mathematics
10 LESSON 4-4 CONTINUED Example 4 A certain cell has a diameter of approximately meters. A second cell has a diameter of meters. Which cell has a greater diameter? Compare powers of Compare the values between 1 and The has a greater diameter. Check It Out! 1. Write the number in standard notation Write in scientific notation. 3. An oil rig can hoist,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. 4. A certain cell has a diameter of approximately meters. A second cell has a diameter of meters. Which cell has a greater diameter? 78 Holt McDougal Mathematics
11 LESSON 4-5 Squares and Square Roots Lesson Objectives Find square roots Vocabulary square root (p. 18) principal square root (p. 18) perfect square (p. 18) Additional Examples Example 1 Find the two square roots of each number. A is a square root, since 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). 79 Holt McDougal Mathematics
12 LESSON 4-5 CONTINUED Example A square window has an area of 169 square inches. How wide is the window? Find the square root of to find the length of the window. Use the square root; a negative square root has no meaning. 169 So 169. The window is inches wide. Example 3 Simplify each expression. A ( ) 7 Evaluate the root. 7 Multiply. B Add Rewrite 5 10 as Evaluate the roots. 4 Add. Check It Out! 1. Find the two square roots of the number Holt McDougal Mathematics
13 LESSON 4-6 Estimating Square Roots Lesson Objectives Estimate square roots and solve problems using square roots Additional Examples Example 1 Each square root is between two consecutive integers. Name the integers. Explain your answer. A. 55 Think: What are squares close to 55? is between and because is between and. B. 90 Think: What are perfect close to 90? ( 9) ( 10) is between and because is between and. 81 Holt McDougal Mathematics
14 LESSON 4-6 CONTINUED Example 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 tenth of an inch. Because 500 is between and 3, the square root of 500 is between and. Guess.5 Guess. Guess.4 Guess Too high Too low Too high Too low Square root is between and.5 Square root is between. and.5 Square root is between. and.4 Square root is between.3 and.4 The square root is between.3 and.4. To round to the nearest tenth, look at the next decimal place. Consider Too The square root must be greater than.35, so round up. To the nearest tenth, 500 is about. The length of each side of the tablecloth is about in. The length of a side of the tablecloth is inches, to the nearest tenth of an inch. Now estimate the length around the tablecloth..4 You will need about Perimeter 4 side inches of fringe. 8 Holt McDougal Mathematics
15 LESSON 4-6 CONTINUED Example 3 Estimate 141 to the nearest hundredth. Step 1: Find the value of the whole number. 141 Find the perfect squares nearest Find the square roots of the perfect squares. 141 The number will be between and. The whole number part of the answer is. Step : 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. Divide to 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. Example 4 Use a calculator to find 600. Round to the nearest tenth. Using a calculator, Rounded, 600 is. 83 Holt McDougal Mathematics
16 LESSON 4-6 CONTINUED Check It Out! The square root is between two consecutive integers. Name the integers You want to build a fence around a square garden that is 50 square feet. Calculate the length of one side of the garden and the total length of the fence, to the nearest tenth. 3. Estimate 89 to the nearest hundredth. 4. Use a calculator to find 800. Round to the nearest tenth. 84 Holt McDougal Mathematics
17 LESSON 4-7 The Real Numbers Lesson Objectives Determine if a number is rational or irrational Vocabulary irrational number (p. 195) real number (p. 195) Density Property (p. 196) Additional Examples 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 Holt McDougal Mathematics
18 LESSON 4-7 CONTINUED Example State if the number is rational, irrational, or not a real number. Justify your answer. A , because is a whole number 3 B. 4 ;because it is the of a negative number C , is rational 3 Example 3 Find a real number between 3 5 and There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by. ( ) A real number between 3 5 and 3 3 is. 5 Check It Out! 1. Write all names that apply to the number. 9. State if the number is rational, irrational, or not a real number Holt McDougal Mathematics
19 LESSON 4-8 The Pythagorean Theorem Lesson Objectives Use the Pythagorean Theorem to solve problems Vocabulary Pythagorean Theorem (p. 00) leg (p. 00) hypotenuse (p. 00) Additional Examples Example 1 Find the length of each hypotenuse to the nearest hundredth. A. c 4 5 a b c c Substitute for a and b. Theorem c Simplify powers. c c Solve for c; c c. c 87 Holt McDougal Mathematics
20 LESSON 4-8 CONTINUED B. triangle with coordinates (1, ), (1, 7), (13, ) 0 y 1 4 x The points form a right triangle with a 9 and b 1. a b c Pythagorean Theorem 0 c Substitute for a and b. c Simplify. c. c Find the square root. Example Solve for the unknown side in the right triangle to the nearest tenth. 5 b a b c b Substitute for a and c. Theorem 7 b Simplify powers. b b Holt McDougal Mathematics
21 LESSON 4-8 CONTINUED 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 75 miles west. How far apart are the two planes after they land? a b c Pythagorean Theorem c Substitute for a and b. c Simplify. c. c Find the square root. The planes are about 805 miles apart after they land. Check It Out! 1. Find the length of the hypotenuse. 5 c 7. Solve for the unknown side in the right triangle. 1 b 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? 4 89 Holt McDougal Mathematics
22 LESSON 4-9 Applying the Pythagorean Theorem and Its Converse Lesson Objectives Use the Distance Formula and the Pythagorean Theorem and its converse to solve problems Additional Examples Example 1 What is the diagonal length of the projector screen? 7 ft 3 ft Find the length of the diagonal of the projector screen. c Use the. c Simplify. c Add. c Take the of both sides. c Find the. The diagonal length is about feet. 90 Holt McDougal Mathematics
23 LESSON 4-9 CONTINUED Example Find the distances between the points to the nearest tenth. 4 J y O 4 L x A. J and K Let J be (x, y ) and K be (x 1, y 1 ). K 4 M d (x x 1 ) (y y 1 ) Use the. ( ) ( ( )) ( ) Substitute. Subtract. Simplify powers. Take the square root. The distance between J and K is units. Example 3 Tell whether the given side lengths form a right triangle. A. 9, 1, 15 a b c Compare a b to c. Substitute. Simplify. Add. The side lengths a right triangle. Check It Out! 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? 91 Holt McDougal Mathematics
24 CHAPTER 4 Chapter Review 4-1 Exponents Simplify ( 6) Write in exponential form q q c 5c 5c 5c 4- Integer Exponents Simplify. Write in decimal form Simplify. 13. ( ) (4 1) ( 1) Properties of Exponents Multiply. Write the product as one power y 10 y a 4 a 3 Divide. Write the quotient as one power p p6 3. r 7 r c c 5 9 Holt McDougal Mathematics
25 CHAPTER 4 REVIEW CONTINUED 4-4 Scientific Notation Write each number in standard notation Write each number in scientific notation. 8. 7,456,000, ,000,000,000, The distance from Earth to Mars is 7,839,000 km. Suppose a rocket traveled from Earth to Mars and back 50 times. How many km did the rocket travel? Write your answer in scientific notation. 4-5 Squares and Square Roots Find the two square roots of each number Evaluate each expression Estimating Square Roots Each square root is between two consecutive integers. Name the integers , Use a calculator to find each value. Round to the nearest tenth Holt McDougal Mathematics
26 CHAPTER 4 REVIEW CONTINUED 4-7 The Real Numbers Write all names that apply to each number State if the number is rational, irrational, or not a real number The Pythagorean Theorem Solve for the unknown side of each right triangle to the nearest tenth a c b Applying the Pythagorean Theorem and Its Converse Tell whether the given side lengths form a right triangle , 10, , 0, , 11, 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? 94 Holt McDougal Mathematics
27 CHAPTER 4 Big Ideas Answer these questions to summarize the important concepts from Chapter 4 in your own words. 1. Explain how to evaluate Explain the difference between and Explain why Explain Explain how to estimate Explain why 0.3 is a rational number. For more review of Chapter 4: Complete the Chapter 4 Study Guide and Review on pages 1 14 of your textbook. Complete the Ready to Go On quizzes on pages 180 and 08 of your textbook. 95 Holt McDougal Mathematics
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