DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Exponents and MODULE Scientific Notation 2 LESSON 2.1 ? ESSENTIAL QUESTION Integer Exponents How can you use scientific notation to solve real-world LESSON 2.2 problems? Scientific Notation with Positive Powers of 10 LESSON 2.3 Scientific Notation with Negative Powers of 10 LESSON 2.4 Operations with Scientific Notation Real-World Video The distance from Earth to other planets, moons, and stars is a very great number of kilometers. To make it easier to write very large and very small my.hrw.com numbers, we use scientific notation. © Houghton © Mifflin Houghton Harcourt • Publishing Company Image ©Eyebyte/ Credits: Alamy Images my.hrw.com my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your Scan with your smart Interactively explore Get immediate write-in student phone to jump directly key concepts to see feedback and help as edition, accessible on to the online edition, how math works. you work through any device. video tutor, and more. practice sets. 29 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Reading Start-Up Vocabulary Review Words Visualize Vocabulary ✔ base (base) ✔ exponent (exponente) Use the ✔ words to complete the Venn diagram. You can put more integers (enteros) than one word in each section of the diagram. ✔ positive number (número positivo) standard notation 2 10 (notación estándar) Preview Words 10 is: 2 is: power (potencia) rational number (número racional) real numbers (número real) scientific notation (notación científica) whole number (número Understand Vocabulary entero) Complete the sentences using the preview words. 1. A number produced by raising a base to an exponent is a . 2. is a method of writing very large or very small numbers by using powers of 10. 3. A is any number that can be expressed as a ratio of two integers. Active Reading Two-Panel Flip Chart Create a two-panel flip chart to help you understand the concepts in this module. Label one flap “Positive Powers of 10” and © Mifflin Houghton Harcourt Publishing Company the other flap “Negative Powers of 10.” As you study each lesson, write important ideas under the appropriate flap. Include sample problems that will help you remember the concepts later when you look back at your notes. 30 Unit 1 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Are YOU Ready? Complete these exercises to review skills you will Personal need for this module. Math Trainer Online Assessment and my.hrw.com Intervention Exponents Write the exponential expression EXAMPLE 10 4 10 10 10 10 = × × × as a product. = 10,000 Simplify. Write each exponential expression as a decimal. 1. 10 2 2. 10 3 3. 10 5 4. 10 7 Multiply and Divide by Powers of 10 5 EXAMPLE 0.0478 × 10 = 0.0478 × 100,000 Identify the number of zeros in the power of 10. 4,780 = When multiplying, move the decimal point to the right the same number of places as the number of zeros. 4 37.9 ÷ 10 = 37.9 ÷ 10,000 Identify the number of zeros in the power of 10. 0.00379 = When dividing, move the decimal point to the left the same number of places as the number of zeros. Find each product or quotient. 3 2 6 4 5. 45.3 × 10 6. 7.08 ÷ 10 7. 0.00235 × 10 8. 3,600 ÷ 10 2 5 4 6 9. 0.5 × 10 10. 67.7 ÷ 10 11. 0.0057 × 10 12. 195 ÷ 10 © Houghton © Mifflin Houghton Harcourt Publishing Company Module 2 31 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Are YOU Ready? (cont'd) Complete these exercises to review skills you will need for this module. Exponents 13. Chris has an MP3 player that can store about 10 8 bytes of data. Explain how to determine the number of zeros in the decimal for 10 8 . Then write the decimal and state the number of bytes in words, without using exponents. 14. Explain how to write 10 5 as a decimal. 4 15. To write 10 as a decimal, Tyrese multiplied 10 × 4 to get 40. Identify Tyrese’s error, and write the correct value. Multiply and Divide by Powers of 10 3 3 16. Explain how to find the product 0.8 × 10 . Compare this product with 10 . 4 17. Jake found that the quotient 9.236 ÷ 10 is 0.0009236. If Jake’s result is correct, justify his answer. If not, give the correct decimal. 5 18. To find the quotient 62.5 ÷ 10 , Gina first identified the number of zeros in the power 5 of 10. When written as a decimal, 10 has five zeros. Describe how Gina can complete © Mifflin Houghton Harcourt Publishing Company the problem. 32 Unit 1 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A LESSON 8.1.2.1 Students will develop and use the properties of integer 2.1 Integer Exponents exponents. ESSENTIAL QUESTION ? How can you develop and use the properties of integer exponents? EXPLORE ACTIVITY 1 Using Patterns of Integer Exponents The table below shows powers of 5, 4, and 3. 4 3 2 1 0 1 2 5 = 625 5 = 125 5 = 25 5 = 5 5 = 5 - = 5 - = 4 3 2 1 0 1 2 4 = 256 4 = 64 4 = 16 4 = 4 4 = 4 - = 4 - = 4 3 2 1 0 1 2 3 = 81 3 = 27 3 = 9 3 = 3 3 = 3 - = 3 - = A What pattern do you see in the powers of 5? B What pattern do you see in the powers of 4? C What pattern do you see in the powers of 3? D Complete the table for the values of 5 0 , 5 -1 , 5 -2 . E Complete the table for the values of 4 0 , 4 -1 , 4 -2 . F Complete the table for the values of 3 0 , 3 -1 , 3 -2 . © Houghton © Mifflin Houghton Harcourt Publishing Company Reflect 1. Make a Conjecture Write a general rule for the value of a0. 2. Make a Conjecture Write a general rule for the value of a-n. Lesson 2.1 33 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A EXPLORE ACTIVITY 2 Exploring Properties of Integer Exponents A Complete the following equations. 3 · 3 · 3 · 3 · 3 = 3 (3 · 3 · 3 · 3) · 3 = 3 · 3 = 3 (3 · 3 · 3) · (3 · 3) = 3 · 3 = 3 What pattern do you see when multiplying two powers with the same base? 2 5 Use your pattern to complete this equation: 5 · 5 = 5 . B Complete the following equation: 4 5 4 4 4 4 4 14 14 14 4 4 __ __________· · · · __________· · · · 3 = 4 4 4 = 4 4 4 = 4 · 4 = 4 4 · · 1· 1· 1 What pattern do you see when dividing two powers with the same base? 8 Math Talk __ 6 Mathematical Processes Use your pattern to complete this equation: 3 = 6 . 6 Do the patterns you found in parts A–C apply if the C Complete the following equations: exponents are negative? If so, give an example of each. ( 3)2 5 = (5 · 5 · 5) = (5 · 5 · 5) · (5 · 5 · 5) = 5 What pattern do you see when raising a power to a power? © Mifflin Houghton Harcourt Publishing Company ( 2)4 Use your pattern to complete this equation: 7 = 7 . 34 Unit 1 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Reflect Let m and n be integers. m n 3. Make a Conjecture Write a general rule for the value of a · a . a m 4. Make a Conjecture Write a general rule for the value of __ , a ≠ 0. a n 5. Make a Conjecture Write a general rule for the value of ( a m ) n . Applying Properties of Integer Exponents You can use the general rules you found in the Explore Activities to simplify more complicated expressions. Math On the Spot my.hrw.com EXAMPLEL 1 My Notes Simplify each expression. 5 8 0 A (5 - 2) · 3 - + (5 + 2) 5 8 0 (3) · 3 - + (7) Simplify within parentheses. 5 ( 8) 3 + - + 1 Use properties of exponents. 3 3 - + 1 Simplify. 1 1 __ __ 27 + 1 = 1 27 Apply the rule for negative exponents and add. 2 3 (3 1) B [________ + ] 2 (7 - 3) ( 4 2 ) 3 ____ Simplify within parentheses. 4 2 4 6 __ Use properties of exponents. 4 2 4 6-2 Use properties of exponents. 4 4 = 256 Simplify. YOUR TURN © Houghton © Mifflin Houghton Harcourt Publishing Company Simplify each expression. 2 2 3 6. [ _______ (6 - 1) ] 7. ( 2 2 ) (10 6) 3 4 -5 3 - - ∙ Personal (3 + 2) Math Trainer Online Assessment and Intervention my.hrw.com Lesson 2.1 35 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Guided Practice Find the value of each power. (Explore Activity 1) 1 2 0 1. 8 - = 2. 6 - = 3. 25 6 = 2 4 5 4. 10 = 5. 5 = 6. 2 - = 5 0 3 7. 4 - = 8. 8 9 = 9. 11 - = Use properties of exponents to write an equivalent expression. (Explore Activity 2) 10. 4 · 4 · 4 = 4 11. (2 · 2) · (2 · 2 · 2) = 2 · 2 = 2 7 12 - __ 6 _______________6 · 6 · 6 · 6 · 6 · 6 · 6 ___ 8 12. 5 13. 9 8 6 = 6 · 6 · 6 · 6 · 6 = 8 = = 10 8 5 14. 5 · 5 · 5 = 5 15. 7 · 7 = 2 4 3 3 3 16. ( 6 ) = (6 · 6) 17. ( 3 ) = (3 · 3 · 3) (6 6) (6 6) (3 3 3) = · · · · · · = · · · · · ( ) ( ) = = Simplify each expression.