1 Lesson 5.2.3 Scientific Notation. 2 Lesson 5.2.3 Scientific Notation California Standard: Number...

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Lesson 5.2.3Lesson 5.2.3

Scientific NotationScientific Notation

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Lesson

5.2.3Scientific NotationScientific Notation

California Standard:Number Sense 1.1Read, 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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Scientific NotationScientific NotationLesson

5.2.3

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 × 107

128,000,000,000 = 1.28 × 1011

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You Can Use Powers of 10 to Write Large Numbers

Lesson

5.2.3

In Chapter 2 you saw how to write large numbers as a product of two factors using scientific notation.

1,200,000 = 1.2 × 106

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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Example 1

Solution follows…

Lesson

5.2.3

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 × 108 miles

Split the number into the appropriate factors.

Write the power of ten in base and exponent form.

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Guided Practice

Solution follows…

Lesson

5.2.3

Write the numbers in Exercises 1–6 in scientific notation.

1. 487,000,000,000 2. 6000

3. 93,840,000 4. –1,630,000,000,000

5. 28,410,000,000,000 6. –3,854,000,000

4.87 × 1011 6 × 103

9.384 × 107 –1.63 × 1012

2.841 × 1013 –3.854 × 109

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You Can Write Small Numbers in Scientific Notation

Lesson

5.2.3

Scientific notation is also a useful way to write small numbers.

5.4 × 10–6 is 0.0000054 written in scientific notation.

Using powers of 10 you can write this as 5.4 ÷ 106.

= 5.4 × 10–6= 5.4 ÷ 1060.0000054 = 5.4 ÷ 1,000,000

A number like 0.0000054 can be rewritten as a division.

And remember that 1 ÷ 106 = = 10–6.1

106

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Example 2

Solution follows…

Lesson

5.2.3

A red blood cell has a diameter of 0.000007 m. Write this number in scientific notation.

Solution

0.000007 = 7 ÷ 1,000,000

= 7 × 10–6 m

= 7 ÷ 106

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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Guided Practice

Solution follows…

Lesson

5.2.3


7. 0.000419 8. 0.000000000015

9. 0.00000007 10. 0.000030024

11. 0.00008946 12. 0.00000004645

4.19 × 10–4 1.5 × 10–11

7 × 10–8 3.0024 × 10–5

8.946 × 10–5 4.645 × 10–8

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You Can Convert Numbers from Scientific Notation

Lesson

5.2.3

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.

12.35 × 10 = 123.5

12.35 ÷ 10 = 1.235

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Example 3

Solution follows…

Lesson

5.2.3

Write 3.0 × 1011 in numeric form.

Solution

= 300,000,000,000

“3.0 × 1011” means “multiply 3.0 by 10, 11 times.”

To multiply 3.0 by 1011, 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 × 1011 = 3.00000000000 × 1011 The green line shows the decimal point moving 11 places to the right.

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Example 4

Solution follows…

Lesson

5.2.3

Write 4.2 × 10–10 in numeric form.

Solution

= 0.00000000042

“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:

00000000004.2 × 10–10

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Guided Practice

Solution follows…

Lesson

5.2.3

In Exercises 13–20, rewrite each number in numerical form.

13. 5.91 × 106 14. 5.91 × 10–6

15. 2.2 × 103 16. 4.85 × 10–8

17. 9.023 × 107 18. 6.006 × 10–2

19. 8.17 × 1010 20. 7.101 × 10–5

5,910,000 0.00000591

2200 0.0000000485

90,230,000 0.06006

81,700,000,000 0.00007101

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Independent Practice

Solution follows…

Lesson

5.2.3


1. 78,000 2. 0.00000091

3. 843,000,000,000 4. 0.00000000000416

5. 20,057,000,000,000

6. 0.000000000000000000000100801

7.8 × 104 9.1 × 10–7

8.43 × 1011 4.16 × 10–12

2.0057 × 1013

1.00801 × 10–22

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Solution follows…

Lesson

5.2.3

Write the numbers in Exercises 7–12 in numerical form.

7. 8.0 × 104 8. 6.2 × 10–5

9. 2.18 × 106 10. 3.03 × 10–10

11. 5.0505 × 109 12. 9.64 × 10–3

80,000 0.000062

2,180,000 0.000000000303

5,050,500,000 0.00964

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Solution follows…

Lesson

5.2.3

13. 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 0.0000158 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 × 109 miles

1.58 × 10–5 miles

0.000000767

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Solution follows…

Lesson

5.2.3

16. An electron's mass is approximately 9.1093826 × 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 × 109 people. Express this population in numeric form. How would you say this number in words?

0.00000000000000000000000000000091093826 kg

$6.9 × 1010

6,100,000,000Six billion, one hundred million

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Round UpRound Up

Lesson

5.2.3

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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