Powers of Ten and Scientific Notation Warm Up Lesson Presentation

Warm Up Find each value. 1. 92 3. 152 5. 103 81 2. 122 144 225 4. 102 100 1,000 6. 104 10,000

Goal Learn to express large numbers in scientific notation.

Vocabulary standard form scientific notation

The distance from Venus to the Sun is over 100,000,000 kilometers The distance from Venus to the Sun is over 100,000,000 kilometers. You can write this number as a power of ten by using a base of ten and an exponent. 10 · 10 · 10 · 10 · 10 · 10 · 10 · 10 = 108 Power of ten

The table shows several powers of ten. Power of 10 Meaning Value 101 10 10 102 10 · 10 100 103 10 · 10 · 10 1,000 104 10 · 10 · 10 · 10 10,000

Additional Example 1: Multiplying by Powers of Ten 14 · 103 = 14 · (10 · 10 · 10) Use 10 as a factor 3 times. = 14 · 1,000 Multiply. = 14,000

Try This: Example 1 Multiply 12 · 102. 12 · 102 = 12 · (10 · 10) Use 10 as a factor 2 times. = 12 · 100 Multiply. = 1,200

You can also find the product of a number and a power of ten simply by moving the decimal point of the number. For powers of ten with positive exponents, move the decimal point to the right.

Additional Example 2: Multiplying by Powers of Ten Mentally Find each product. A. 212 · 104 4 places 212 · 104 = 212.0000 Move the decimal point 4 places. = 2,120,000 (You will need to add 4 zeros.) B. 31.6 · 103 3 places 31.6 · 103 = 31.600 Move the decimal point 3 places. = 31,600 (You will need to add 2 zeros.)

Insert Lesson Title Here Try This: Example 2 Find each product. A. 22.5 · 102 2 places 22.5 · 102 = 22.50 Move the decimal point 2 places. = 2,250 (You will need to add 1 zero.) B. 39.5 · 103 3 places 39.5 · 103 = 39.500 Move the decimal point 3 places. = 39,500 (You will need to add 2 zeros.)

Numbers are usually written in standard form Numbers are usually written in standard form. For example, 17,900,000 is in standard form. Scientific notation is a kind of shorthand that can be used to write large numbers. Numbers expressed in scientific notation are written as the product of two factors. In scientific notation, 17,900,000 is written as 7 1.79 x 10 A number greater than or equal to 1 but less than 10 A power of 10

Writing Math In scientific notation, it is customary to use a multiplication cross () instead of a dot.

Additional Example 3A: Writing Numbers in Scientific Notation Write the number in scientific notation. A. 4,340,000 6 places Move the decimal point to get a number that is greater than or equal to 1 and less than 10. 4,340,000 = 4,340,000 The exponent is equal to the number of places the decimal point is moved. = 4.34  106

Additional Example 3B: Writing Numbers in Scientific Notation Write the number in scientific notation. B. 327,000,000 8 places Move the decimal point to get a number that is greater than or equal to 1 and less than 10. 327,000,000 = 327,000,000 The exponent is equal to the number of places the decimal point is moved. = 3.27  108

Insert Lesson Title Here Try This: Example 3A Write the number in scientific notation. A. 8,421,000 6 places Move the decimal point to get a number that is greater than or equal to 1 and less than 10. 8,421,000 = 8,421,000 The exponent is equal to the number of places the decimal point is moved. = 8.421  106

Insert Lesson Title Here Try This: Example 3B Write the number in scientific notation. B. 327,000 5 places Move the decimal point to get a number that is greater than or equal to 1 and less than 10. 327,000 = 327,000 The exponent is equal to the number of places the decimal point is moved. = 3.27  105

Additional Example 4: Writing Numbers in Standard Form The population of China in the year 2000 was estimated to be about 1.262  109. Write this number in standard form. Since the exponent is 9, move the decimal point 9 places to the right. 1.262  109 = 1.262000000 = 1,262,000,000 The population of China was about 1,262,000,000 people.

Insert Lesson Title Here Try This: Example 4 The distance from the Earth to the Sun is calculated to be 1.5  108 kilometers. Write this distance in standard form. Since the exponent is 8, move the decimal point 8 places to the right. 1.5  108 = 1.50000000 = 150,000,000 The distance from the Earth to the Sun is about 150,000,000 kilometers.

Insert Lesson Title Here Lesson Quiz: Part 1 Multiply. 1. 25  102 2,500 2. 18  104 180,000 Find each product. 3. 110  102 11,000 4. 3.742  103 3,742

Insert Lesson Title Here Lesson Quiz: Part 2 Write each number in scientific notation.. 5. 7,400,000 6. 45,000 7. Earth is about 9.292  107 miles from the Sun. Write this number in standard form. 7.4  106 4.5 104 92,920,000