Operations in Scientific Notation

Transcription

1 .7 Operations in Scientific Notation How can you perform operations with numbers written in scientific notation? 1 ACTIVITY: Adding Numbers in Scientific Notation Work with a partner. Consider the numbers 2. 3 and a. Explain how to use order of operations to find the sum of these numbers. Then find the sum b. The factor is common to both numbers. How can you use the Distributive Property to rewrite the sum ( 2. 3 ) + ( )? ( 2. 3 ) + ( ) = Distributive Property c. Use order of operations to evaluate the expression you wrote in part (b). Compare the result with your answer in part (a). d. STRUCTURE Write a rule you can use to add numbers written in scientific notation where the powers of are the same. Then test your rule using the sums below. (.9 5 ) + ( ) = ( 3.85 ) + ( 5.72 ) = 2 ACTIVITY: Adding Numbers in Scientific Notation Work with a partner. Consider the numbers 2. 3 and 7.1. Scientific Notation In this lesson, you will add, subtract, multiply, and divide numbers written in scientific notation. a. Explain how to use order of operations to find the sum of these numbers. Then find the sum b. How is this pair of numbers different from the pairs of numbers in Activity 1? c. Explain why you cannot immediately use the rule you wrote in Activity 1(d) to find this sum. d. STRUCTURE How can you rewrite one of the numbers so that you can use the rule you wrote in Activity 1(d)? Rewrite one of the numbers. Then find the sum using your rule and compare the result with your answer in part (a). e. REASONING Do these procedures work when subtracting numbers written in scientific notation? Justify your answer by evaluating the differences below. ( ) (.6 5 ) = ( ) ( 1.5 ) = 8 Chapter Exponents and Scientific Notation

2 Math Practice Justify Conclusions Which step of the procedure would be affected if the powers of were different? Explain. 3 ACTIVITY: Multiplying Numbers in Scientific Notation Work with a partner. Match each step with the correct description. Step Description ( 2. 3 ) ( ) Original expression 1. = A. Write in standard form. 2. = (2. 7.1) ( 3 3 ) B. Product of Powers Property 3. = C. Write in scientific notation.. = D. Commutative Property of Multiplication 5. = E. Simplify. 6. = 17,00,000 F. Associative Property of Multiplication Does this procedure work when the numbers have different powers of? Justify your answer by using this procedure to evaluate the products below. ( ) ( ) = ( 8. 6 ) ( 5.7 ) = ACTIVITY: Using Scientific Notation to Estimate Work with a partner. A person normally breathes about 6 liters of air per minute. The life expectancy of a person in the United States at birth is about 80 years. Use scientific notation to estimate the total amount of air a person born in the United States breathes over a lifetime. 5. IN YOUR OWN WORDS How can you perform operations with numbers written in scientific notation? 6. Use a calculator to evaluate the expression. Write your answer in scientific notation and in standard form. a. ( 1.5 ) + ( 6.3 ) b. ( ) ( ) c. (.1 3 ) (.3 3 ) d. (.75 6 ) ( ) Use what you learned about evaluating expressions involving scientific notation to complete Exercises 3 6 on page 52. Section.7 Operations in Scientific Notation 9

3 .7 Lesson Lesson Tutorials To add or subtract numbers written in scientific notation with the same power of, add or subtract the factors. When the numbers have different powers of, first rewrite the numbers so they have the same power of. Study Tip EXAMPLE In Example 1(b), you will get the same answer when you start by rewriting as Adding and Subtracting Numbers in Scientific Notation Find the sum or difference. Write your answer in scientific notation. a. (.6 3 ) + ( ) = ( ) 3 Distributive Property = Add. = ( ) 3 Write in scientific notation. = Product of Powers Property b. ( ) ( ) Rewrite so that it has the same power of as = Rewrite 3 as 1 2. = Rewrite as Subtract the factors. ( ) ( ) = ( ) 2 Distributive Property = Subtract. Exercises 7 1 Find the sum or difference. Write your answer in scientific notation. 1. ( ) + ( ) 2. ( ) (.5 5 ) To multiply or divide numbers written in scientific notation, multiply or divide the factors and powers of separately. Study Tip EXAMPLE You can check your answer using standard form. ( 3 5 ) ( 5 2 ) = = = Multiplying Numbers in Scientific Notation Find ( 3 5 ) ( 5 2 ). Write your answer in scientific notation. ( 3 5 ) ( 5 2 ) = Commutative Property of Multiplication = (3 5) ( 5 2 ) Associative Property of Multiplication = 15 7 Simplify. = Write 15 in scientific notation. = Product of Powers Property 50 Chapter Exponents and Scientific Notation

4 EXAMPLE 3 Dividing Numbers in Scientific Notation Find 7. Write your answer in scientific notation = Rewrite as a product of fractions. = Divide 1.5 by 6. = Quotient of Powers Property = Write 0.25 in scientific notation. = Product of Powers Property Exercises Find the product or quotient. Write your answer in scientific notation ( 8 5 ). ( 7 2 ) ( 3 5 ) 5. ( ).6 6. ( ) ( ) EXAMPLE Diameter 1,00,000 km Real-Life Application How many times greater is the diameter of the Sun than the diameter of Earth? Write the diameter of the Sun in scientific notation. 1,00,000 = Divide the diameter of the Sun by the diameter of Earth = Rewrite as a product of fractions. = Divide 1. by Diameter 1.28 km = Quotient of Powers Property = Write in standard form. The diameter of the Sun is about 9 times greater than the diameter of Earth. 7. How many more kilometers is the radius of the Sun than the radius of Earth? Write your answer in standard form. Section.7 Operations in Scientific Notation 51

5 .7 Exercises Help with Homework 1. WRITING Describe how to subtract two numbers written in scientific notation with the same power of. 2. NUMBER SENSE You are multiplying two numbers written in scientific notation with different powers of. Do you have to rewrite the numbers so they have the same power of before multiplying? Explain. 9+(-6)=3 3+(-3)= +(-9)= 9+(-1)= Evaluate the expression using two different methods. Write your answer in scientific notation. 3. ( ) + ( ). ( ) + ( 3. 5 ) 5. ( ) ( ) 6. (.5 ) ( ) 1 Find the sum or difference. Write your answer in scientific notation. 7. ( 2 5 ) + ( ) 8. ( ) (.5 9 ) 9. ( ) ( 8 ). ( ) + ( 5. 6 ) 11. ( ) ( ) 12. ( 5 5 ) + ( ) 13. ( ) + ( ) 1. ( 2. 1 ) ( ) 15. ERROR ANALYSIS Describe and correct the error in finding the sum of the numbers. ( ) + ( ) = ( ) ( 9 8 ) = Find the product or quotient. Write your answer in scientific notation ( 7 7 ) 17. ( ) ( 2 3 ) 18. ( ) 19. ( 5 7 ) ( 3 6 ) 20. ( ) ( ) 21. ( ) ( 7 ) 22. ( ) ( 1. 5 ) 23. ( 2.8 ) ( ) 2. MONEY How many times greater is the thickness of a dime than the thickness of a dollar bill? Thickness cm Thickness cm 52 Chapter Exponents and Scientific Notation

6 Evaluate the expression. Write your answer in scientific notation ,200,000 ( ) ( ) 26. ( 9 3 ) + ( 2. 5 ) GEOMETRY Find the perimeter of the rectangle. Area cm BLOOD SUPPLY A human heart pumps about 7 2 liter of blood per heartbeat. The average human heart beats about 72 times per minute. How many liters of blood does a heart pump in 1 year? in 70 years? Write your answers in scientific notation. Then use estimation to justify your answers cm Not drawn to scale cm cm.26 cm 29. DVDS On a DVD, information is stored on bumps that spiral around the disk. There are 73,000 ridges (with bumps) and 73,000 valleys (without bumps) across the diameter of the DVD. What is the diameter of the DVD in centimeters? 30. PROJECT Use the Internet or some other reference to find the populations and areas (in square miles) of India, China, Argentina, the United States, and Egypt. Round each population to the nearest million and each area to the nearest thousand square miles. a. Write each population and area in scientific notation. b. Use your answers to part (a) to find and order the population densities (people per square mile) of each country from least to greatest. 31. Albert Einstein s most famous equation is E = mc 2, where E is the energy of an object (in joules), m is the mass of an object (in kilograms), and c is the speed of light (in meters per second). A hydrogen atom has joule of energy and a mass of kilogram. What is the speed of light? Write your answer in scientific notation. Find the cube root. (Section 7.2) MULTIPLE CHOICE What is the volume of the cone? (Section 8.2) A 16π cm 3 B 8π cm 3 C 8π cm 3 D 1π cm cm 9 cm Section.7 Operations in Scientific Notation 53