Scientific Notation Main IDEA More than 425 million pounds of gold has been discovered in the world. If all this gold were in one place, it would form a cube seven stories on each side. Express numbers in scientific notation and in standard form. New Vocabulary 1. Write 425 million in standard form. scientific notation 2. Complete: 4.25 × ? = 425 million. Math Online glencoe.com When you deal with very large numbers like 425,000,000, it can be difficult to keep track of the zeros. You can express numbers such as this in scientific notation by writing the number as the product of a factor and a power of 10. •Personal Tutor •Self-Check Quiz •Extra Examples Scientific Notation Key Concept Words A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. Symbols a × 10 n, where 1 ≤ a < 10 and n is an integer Examples 425,000,000 = 4.25 × 10 8 Express Large Numbers in Standard Form 1 Express 2.16 × 10 5in standard form. 2.16 × 1 0 5= 2.16 × 100,00010 5= 100,000 = 216,000 Move the decimal point 5 places to the right. Express each number in standard form. a. 7.6 × 10 6 b. 3.201 × 10 4 Scientific notation is also used to express very small numbers. Study the pattern of products at the right. Notice that multiplying by a negative power of 10 moves the decimal point to the left the same number of places as the absolute value of the exponent. LA2 Looking Ahead 1.25 × 10 2= 125 1.25 × 10 1= 12.5 1.25 × 10 0= 1.25 1.25 × 10 –1= 0.125 1.25 × 10 –2= 0.0125 1.25 × 10 –3= 0.00125 Express Small Numbers in Standard Form 2 Express 5.8 × 1 0 –3in standard form. Move the Decimal Point The exponent tells you how many places to move the decimal point. • If the exponent is 5.8 × 10 –3= 5.8 × 0.001 10–3 = 0.001 positive, move the decimal point to the right. • If the exponent is negative, move the decimal point to the left. = 0.0058 c. 4.7 × 10 –5 Move the decimal point 3 places to the left. d. 9.0 × 10 –4 To write a number in scientific notation, place the decimal point after the first nonzero digit. Then find the power of 10. Express Numbers in Scientific Notation Express each number in scientific notation. 3 1,457,000 1,457,000= 1.457 × 1,000,000 The decimal point moves 6 places to the left. = 1.457 × 10 6 The exponent is positive. 4 0.00063 0.00063= 6.3 × 0.0001 The decimal point moves 4 places to the right. = 6.3 × 10 –4 The exponent is negative. e. 35,000 f. 0.00722 To compare numbers in scientific notation, first compare the exponents. With positive numbers, any number with a greater exponent is greater. If the exponents are the same, compare the factors. Compare Numbers in Scientific Notation 5 OCEANS The Atlantic Ocean has an area of 3.18 × 10 7square miles. The Pacific Ocean has an area of 6.4 × 10 7square miles. Which ocean has the greater area? Since both exponents are the same, compare the factors. 3.18 < 6.4 → 3.18 × 10 7< 6.4 × 10 7 So, the Pacific Ocean has the greater area. g. Replace ● with <, >, or = to make 4.13 × 10 –2● 5.0 × 10 –3a true sentence. Lesson 1 Scientific Notation LA3 Examples 1, 2 (pp. LA2–LA3) Express each number in standard form. 1. 3.754 × 10 5 3. 1.5 × 10 –4 Examples 3, 4 (p. LA3) 2. 8.34 × 10 6 4. 2.68 × 10 –3 Express each number in scientific notation. 5. 4,510,000 6. 0.00673 7. 0.000092 8. 11,620,000 9. Physical science Light travels 300,000 kilometers per second. Write this number in scientific notation. Example 5 (p. LA3) 10. Technology The distance between tracks on a CD and DVD are shown in the table. Which disc has the greater distance between tracks? Disc CD DVD Distance (mm) 1.6 × 10 -3 7.4 × 10 -4 Replace each ● with <, >, or = to make a true sentence. 11. 2.3 × 10 5● 1.7 × 10 5 HOMEWORK For Exercises 13–22 23–32 33–38 HELP See Examples 1, 2 3, 4 5 12. 0.012 ● 1.4 × 10 –1 Express each number in standard form. 13. 6.1 × 10 4 14. 5.72 × 10 6 15. 3.3 × 10 –1 16. 5.68 × 10 –3 17. 9.014 × 10 –2 18. 1.399 × 10 5 19. 2.505 × 10 3 20. 7.4 × 10 –5 21. SPIDERS The diameter of a spider’s thread is 1.0 × 1 0 –3inch. Write this number in standard form. 22. dinosaurs The Gigantosaurus dinosaur weighed about 1.4 × 1 0 4pounds. Write this number in standard form. Express each number in scientific notation. 23. 499,000 24. 2,000,000 25. 0.006 26. 0.0125 27. 50,000,000 28. 39,560 29. 0.000078 30. 0.000425 31. CHESS The number of possible ways that a player can play the first four moves in a chess game is 3 billion. Write this number in scientific notation. 32. SCIENCE A particular parasite is approximately 0.025 inch long. Write this number in scientific notation. SPORTS For Exercises 33 and 34, use the table. Determine which category in each pair had a greater amount of sales. 33. golf or tennis LA4 Looking Ahead 34. camping or golf Category Camping Golf Tennis Sales ($) 1.547 × 10 9 3.243 × 10 9 3.73 × 10 8 Source: National Sporting Goods Assoc. Replace each ● with <, >, or = to make a true sentence. 35. 1.8 × 10 3● 1.9 × 10 –1 37. 0.00701 ● 7.1 × 10 –3 39. measurement The table at the right shows the values 36. 5.2 × 10 2● 5000 38. 6.49 × 10 4● 649 × 10 2 of different prefixes that are used in the metric system. Write the units attometer, gigameter, kilometer, nanometer, petameter, and picometer in order from greatest to least measure. 40. NUMBER SENSE Write the product of 0.00004 and 0.0008 in scientific notation. Metric Measures Prefix Meaning atto 10 -18 giga 10 9 kilo 10 3 nano 10 -9 peta 10 15 pico 10 -12 41. NUMBER SENSE Order 6.1 × 1 0 4, 6100, 6.1 × 10 –5, 0.0061, and 6.1 × 10 –2 from least to greatest. PHYSICAL SCIENCE For Exercises 42 and 43, use the table. The table shows the maximum amounts of lava in cubic meters per second that erupted from four volcanoes. 42. How many times greater was the Mount St. Helens eruption than the Ngauruhoe eruption? Volcanic Eruptions Eruption Rate Volcano, Year (m 3/s) Mount St. Helens, 2.0 × 10 4 1980 Ngauruhoe, 1975 2.0 × 10 3 Hekla, 1970 4.0 × 10 3 Agung, 1963 3.0 × 10 4 Source: University of Alaska 43. How many times greater was the Hekla eruption than the Ngauruhoe eruption? Write each number in standard form. 44. (8 × 10 0) + (4 × 10 –3) + (3 × 10 –5) 45. (4 × 10 4) + (8 × 10 3) + (3 × 10 2) + (9 × 10 1) + (6 × 10 0) H.O.T. Problems 46. Challenge Convert the numbers in each expression to scientific notation. Then evaluate the expression. Express in scientific notation and in decimal notation. (420,000)(0.015) 0.025 a. __ (0.078)(8.5) 0.16(250,000) b. __ 47. REASONING Which is a better estimate for the number of times per year 48. Open Ended Describe a real-life value or measure using numbers in that a person blinks: 6.25 × 10 –2times or 6.25 × 10 6times? Explain your reasoning. scientific notation and in standard form. 49. WR ITING IN MATH Explain the relationship between a number in standard form and the sign of the exponent when the number is written in scientific notation. Lesson 1 Scientific Notation LA5