Domain 2 Lesson 7 Scientific Notation Common Core Standard: 8.EE.4 Getting the Idea Scientific notation is a way to abbreviate very large or very small numbers using powers of 10. A number written in scientific notation consists of two factors. The first factor is a number greater than or equal to 1, but less than 10. The second factor is a power of 10. Here are some guidelines and examples of numbers written in scientific notation. Standard Form Scientific Notation Numbers $ 10 8,000,000 8 3 10 6 Numbers $ 1 and, 10 3 3 3 10 0 Numbers. 0 and, 1 0.0007 7 3 10 24 Recall that a number raised to the power of 0 equals 1, so multiplying by 10 0 is the same as multiplying by 1. Example 1 Jupiter s minimum distance from the sun is about 460,100,000 miles. What is that number written in scientific notation? Use the definition of scientific notation to find the two factors. Write the first factor, which must be greater than or equal to 1 and less than 10. Put the decimal point after the first nonzero digit, starting at the left. Drop all zeros after the last nonzero digit. 4.60100000 The first factor is 4.601. Step 2 Find the exponent for the power of 10. Count the number of places that the decimal point was moved. 4.60100000 The decimal point was moved 8 places to the left. Since the original number is greater than 10, the exponent will be positive. 8 is the exponent for the power of 10. 66
Step 4 Solution Write the second factor. The exponent is positive 8. The second factor is 10 8. Write the number in scientific notation. 4.601 3 10 8 Jupiter s minimum distance from the sun is about 4.601 3 10 8 miles. Example 2 Mr. Kendall measured a specimen that was 0.00000045 millimeter long. What is the specimen s length written in scientific notation? Use the definition of scientific notation to find the two factors. Write the first factor, which must be greater than or equal to 1 and less than 10. Put the decimal point after the first nonzero digit, starting from the left. Drop the zeros that precede that digit. 0.000000 45 The first factor is 4.5. Step 2 Find the exponent for the power of 10. Count the number of places that the decimal point was moved. 0.00000045 Step 4 Solution The decimal point was moved 7 places to the right. Since the original number is less than 1, the exponent will be negative. 27 is the exponent for the power of 10. Write the second factor. The exponent is negative 7. The second factor is 10 27. Write the number in scientific notation. 4.5 3 10 27 The specimen was 4.5 3 10 27 millimeter long. 67
When converting from scientific notation to standard form, move the decimal point to the right for a positive power of 10 and to the left for a negative power of 10. Example 3 What is 3.5 3 10 26 written in standard form? Step 2 Look at the exponent of the second factor to move the decimal point. Look at the exponent of the second factor. The exponent is negative, so the decimal point will move to the left. The exponent is 26, so move the decimal point 6 places to the left. Move the decimal point in 3.5 six places to the left. Add zeros as needed. 0.0000035 Use a scientific calculator to check your solution. To find 3.5 3 10 26 : Type 3.5. Press the multiplication sign key. Type 10. Press the exponent key. Type 6. Press the positive/negative key to change the sign on the 6. 3.5 10 ^ 6 Press the equal sign key. The screen should show 0.0000035. The solution is correct. Solution 3.5 3 10 26 5 0.0000035 To multiply numbers in scientific notation, first multiply the decimal factors and then multiply the power-of-10 factors. Remember to use the properties of powers when you multiply the power-of-10 factors. In the example below, a and b are the decimal factors. (a 3 10 m )(b 3 10 n ) 5 ab 3 10 m 1 n 68 Domain 2: Expressions and Equations
Lesson 7: Scientific Notation To divide numbers in scientific notation, first divide the decimal factors. Then divide the power-of-10 factors, using the properties of powers. In the example below, a and b are the decimal factors and b fi 0. (a 3 10 m ) (b 3 10 n ) 5 a b 3 10 m 2 n When you multiply or divide numbers in scientific notation, your product or quotient may not be in scientific notation because the decimal factor is not greater than or equal to 1 and less than 10. To fix this, write the decimal factor in scientific notation and use the properties of powers to simplify the expression. Example 4 Find the product in scientific notation. (1.5 3 10 3 )(7.8 3 10 27 ) Step 2 Multiply the decimal-number factors. Then multiply the power-of-10 factors. Use the commutative and associative properties to regroup the factors. (1.5 3 10 3 )(7.8 3 10 27 ) 5 (1.5 3 7.8)( 10 3 3 10 27 ) Multiply the decimal factors. 1.5 3 7.8 5 11.7 Multiply the power-of-10 factors. 10 3 3 10 27 5 10 3 1 (27) 5 10 24 Step 4 Write the product using the products from Steps 2 and 3. (1.5 3 10 3 )(7.8 3 10 27 ) 5 11.7 3 10 24 Step 5 Solution Write 11.7 3 10 24 in scientific notation. First, write 11.7 in scientific notation. Move the decimal one place to the left, then multiply the result by 10 1. 11.7 written in scientific notation is 1.17 3 10 1. Now, substitute this into the expression and simplify. 11.7 3 10 24 5 1.17 3 10 1 3 10 24 1.17 3 10 1 3 10 24 5 1.17 3 ( 10 1 3 10 24 ) Use the associative property of multiplication. 1.17 3 ( 10 1 3 10 24 ) 5 1.17 3 10 1 1 (24) Use the product of powers property. 1.17 3 10 1 1 (24) 5 1.17 3 10 23 (1.5 3 10 3 )(7.8 3 10 27 ) 5 1.17 3 10 23 69
Example 5 What is 4.2 3 (2.5 3 10 26 ) written in standard form? Step 2 Step 4 Use the associative property to regroup the factors. Then write the product in standard form. Use the associative property to regroup the factors. 4.2 3 (2.5 3 10 26 ) 5 (4.2 3 2.5) 3 10 26 Multiply the decimal factors. 4.2 3 2.5 5 10.5 Rewrite the expression using the result from Step 2 and the power-of-10 factor. (4.2 3 2.5) 3 10 26 5 10.5 3 10 26 Write the product in standard form. Look at the power-of-10 factor. The negative exponent means you move the decimal point to the left. So, 26 means you move the decimal point in 10.5 six places to the left. 0.0000105 Solution 4.2 3 (2.5 3 10 26 ) 5 0.0000105 Example 6 Find the quotient in scientific notation. 8.82 3 10 5 3.6 3 10 3 Divide the decimal-number factors and divide the power-of-10 factors. Step 2 Rewrite the expression. 8.82 3 10 5 5 8.82 3.6 3 10 3 3.6 3 10 5 10 3 Divide the decimal-number factors. 8.82 5 2.45 3.6 Divide the power-of-10 factors. 10 5 10 3 5 10 5 2 3 5 10 2 Step 4 Write the result using the quotients from Steps 2 and 3. 2.45 3 10 2 70 Domain 2: Expressions and Equations
Lesson 7: Scientific Notation Step 5 Solution Check that the product is written in scientific notation. The first factor, 2.45, is greater than or equal to 1 and less than 10. The second factor, 10 2, is a power of 10. The product is written in scientific notation. 8.82 3 10 5 3.6 3 10 3 5 2.45 3 10 2 Coached Example In 2008, the Hartsfield-Jackson Atlanta International Airport ranked as the world s busiest airport. In that year, approximately 9.0 3 10 7 passengers passed through this airport. What is that number written in standard form? Since the exponent is positive, this is a number greater than. The exponent of the second factor is. The exponent tells you to move the decimal point in 9.0 places to the. The number 9.0 3 10 7 in standard form is. About passengers passed through the Hartsfield-Jackson Atlanta International Airport in 2008. 71
Lesson Practice Choose the correct answer. 1. What is 0.000058 written in scientific notation? A. 5.8 3 10 26 B. 5.8 3 10 25 C. 5.8 3 10 5 D. 5.8 3 10 6 2. The length of the Amazon River in South America is 6,400 kilometers. What is this length written in scientific notation? A. 6.4 3 10 2 km B. 6.4 3 10 3 km C. 6.4 3 10 4 km D. 6.4 3 10 5 km 4. The area of Australia is approximately 7,700,000 square kilometers. What is this area written in scientific notation? A. 7.7 3 10 26 sq km B. 7.7 3 10 25 sq km C. 7.7 3 10 5 sq km D. 7.7 3 10 6 sq km 5. What is 4.01 3 10 0 written in standard form? A. 0.401 B. 4.001 C. 4.01 D. 40.1 3. What is 6.92 3 10 23 written in standard form? A. 0.000692 B. 0.00692 C. 0.0692 D. 0.692 6. A virus is viewed under a microscope. Its diameter is 0.0000002 meter. How would this length be expressed in scientific notation? A. 2 3 10 27 meter B. 2 3 10 26 meter C. 2 3 10 6 meters D. 2 3 10 7 meters 72 Domain 2: Expressions and Equations
Lesson 7: Scientific Notation 7. Find the product. (1.9 3 10 3 )(4.5 3 10 2 ) A. 8.55 3 10 1 B. 8.55 3 10 3 C. 8.55 3 10 5 D. 8.55 3 10 6 8. Find the quotient. 2.89 3 10 2 3.4 3 10 22 A. 0.85 3 10 0 B. 0.85 3 10 4 C. 8.5 3 10 3 D. 8.5 3 10 5 9. Mohammed copied this problem into his notebook. (3.4 3 10 5 )(3.8 3 10 29 ) A. Use the associative and commutative properties to rearrange the factors. B. Find the product. Write the product in standard form. 10. Select True or False for each equation. A. 250,000 5 2.5 3 10 5 True False B. 1.4 3 10 22 5 0.014 True False C. 7.85 3 10 22 5 0.00785 True False D. 31,400,000 5 3.14 3 10 7 True False 73
11. For each situation, circle the number in scientific notation that represents the number in standard form. Rama measured the diameter of a molecule as 0.0000035 millimeter. 3.5 10 6 The diameter, in millimeters, of the molecule is 0.35 10 5 35 10 7. The distance from San Francisco, California, to London, England, is about 8,600 kilometers. 8.6 10 2 The distance, in kilometers, from San Francisco to London is about 8.6 10 3 8.6 10 4. 12. Draw a line from each number in scientific notation to its equivalent in standard form. A. 4.5 10 9 45,000 B. 4.5 10 5 0.000045 C. D. E. 4.5 10 3 4.5 10 4 4.5 10 4 4,500,000,000 0.00045 4,500 74 Domain 2: Expressions and Equations
Lesson 7: Scientific Notation 13. Look at each number. Is it equivalent to 3.22 3 10 5? Select Yes or No. A. 0.0000322 Yes No B. (1.4 3 10 2 )(2.3 3 10 3 ) Yes No C. 322,000 Yes No D. (1.4 3 10 1 ) (2.3 3 10 2 ) Yes No 14. Circle every number that is equivalent to 0.0034. A. (4.08 3 1 0 24 ) (1.2 3 10 21 ) B. 3.4 3 10 23 C. (1.7 3 10 2 )(1.2 3 1 0 22 ) D. 3.4 3 10 3 15. Find the standard form equivalent of each number in scientific notation or expression shown below. Write the number or expression in the correct box. (1.7 10 4 )(1.0 10 4 ) (1.7 10 2 )(1.0 10 3 ) 1.7 10 8 1.7 10 5 170,000,000 0.000017 75