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1 Lesson Scientific Notation

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2 Lesson Scientific Notation California Standard: Number Sense 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general. What it means for you: You’ll see how you can use powers of 10 to make very big or very small numbers easier to work with. Key words: scientific notation numeric form power decimal base exponent

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3 Scientific Notation Lesson Scientific notation is a handy way of writing very large and very small numbers. In an earlier lesson, you practiced using powers of ten to write out large numbers. In this Lesson, you’ll get a reminder of how to do that. Then you’ll see that with negative powers, you can do the same thing for very small numbers. 57,000,000 = 5.7 × ,000,000,000 = 1.28 × 10 11

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4 Scientific Notation You Can Use Powers of 10 to Write Large Numbers Lesson In Chapter 2 you saw how to write large numbers as a product of two factors using scientific notation. 1,200,000 = 1.2 × 10 6 The first factor is a number that is at least 1 but less than 10. The second factor is a power of ten. The exponent tells you how many places to move the decimal point to get the number.

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5 Scientific Notation Example 1 Solution follows… Lesson The planet Saturn is about 880,000,000 miles away from the Sun. Write this number in scientific notation. Solution 880,000,000 = 8.8 × 100,000,000 = 8.8 × 10 8 miles Split the number into the appropriate factors. Write the power of ten in base and exponent form.

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6 Scientific Notation Guided Practice Solution follows… Lesson Write the numbers in Exercises 1–6 in scientific notation ,000,000, ,840,0004. –1,630,000,000, ,410,000,000,0006. –3,854,000, × × × 10 7 –1.63 × × –3.854 × 10 9

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7 Scientific Notation You Can Write Small Numbers in Scientific Notation Lesson Scientific notation is also a useful way to write small numbers. 5.4 × 10 –6 is written in scientific notation. Using powers of 10 you can write this as 5.4 ÷ = 5.4 × 10 –6 = 5.4 ÷ = 5.4 ÷ 1,000,000 A number like can be rewritten as a division. And remember that 1 ÷ 10 6 = = 10 –

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8 Scientific Notation Example 2 Solution follows… Lesson A red blood cell has a diameter of m. Write this number in scientific notation. Solution = 7 ÷ 1,000,000 = 7 × 10 –6 m = 7 ÷ 10 6 Split the number into a decimal and a power of ten. Write the power of ten in base and exponent form. Change division by a positive power to multiplication by a negative power.

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9 Scientific Notation Guided Practice Solution follows… Lesson Write the numbers in Exercises 7–12 in scientific notation × 10 –4 1.5 × 10 –11 7 × 10 – × 10 – × 10 – × 10 –8

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10 Scientific Notation You Can Convert Numbers from Scientific Notation Lesson Sometimes you might need to take a number that’s in scientific notation, and write it as an ordinary number. You can use these facts to convert a number from scientific notation back to numeric form. When you divide by 10, the decimal point moves one place to the left. When you multiply by 10, the decimal point moves one place to the right × 10 = ÷ 10 = 1.235

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11 Scientific Notation Example 3 Solution follows… Lesson Write 3.0 × in numeric form. Solution = 300,000,000,000 “3.0 × ” means “multiply 3.0 by 10, 11 times.” To multiply 3.0 by 10 11, all you need to do is move the decimal point 11 places to the right. It might help to write out the 3.0 with extra 0s — then you can see how the decimal point is moving. 3.0 × = × The green line shows the decimal point moving 11 places to the right..

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12 Scientific Notation Example 4 Solution follows… Lesson Write 4.2 × 10 –10 in numeric form. Solution = “4.2 × 10 –10 ” means “divide 4.2 by 10, 10 times.” You need to move the decimal point 10 places to the left. You can write in extra 0s in front of the 4 to help you: × 10 –10

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13 Scientific Notation Guided Practice Solution follows… Lesson In Exercises 13–20, rewrite each number in numerical form × × 10 – × × 10 – × × 10 – × × 10 –5 5,910, ,230, ,700,000,

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14 Scientific Notation Independent Practice Solution follows… Lesson Write the numbers in Exercises 1–6 in scientific notation , ,000,000, ,057,000,000, × × 10 – × × 10 – × × 10 –22

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15 Scientific Notation Independent Practice Solution follows… Lesson Write the numbers in Exercises 7–12 in numerical form × × 10 – × × 10 – × × 10 –3 80, ,180, ,050,500,

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16 Scientific Notation Independent Practice Solution follows… Lesson The planet Uranus is approximately 1,800,000,000 miles away from the Sun. What is this distance in scientific notation? 14. An inch is approximately equal to miles. Write this distance in scientific notation. 15. The volume of the Earth is approximately 7.67 × 10 –7 times the volume of the Sun. Express this figure in numeric form. 1.8 × 10 9 miles 1.58 × 10 –5 miles

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17 Scientific Notation Independent Practice Solution follows… Lesson An electron's mass is approximately × 10 –31 kg. What is this mass in numeric form? 17. In 2006, Congress approved a 69 billion dollar tax cut. What is 69 billion dollars written in scientific notation? 18. At the end of the 20th century, the world population was approximately 6.1 × 10 9 people. Express this population in numeric form. How would you say this number in words? kg $6.9 × ,100,000,000 Six billion, one hundred million

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18 Scientific Notation Round Up Lesson Scientific notation is an important real-life use for powers — it’s called scientific notation because scientists use it all the time to save them having to write out really long numbers.

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