Scientific Notation

Scientific Notation

Powers of 10

Warm UpEvaluate each expression.

1. 123 1,000

2. 123 1,000

3. 0.003 100

4. 0.003 100

5. 104

6. 10–4

7. 230

123,0000.123

0.3

0.00003

10,000

0.0001

1

Evaluate and multiply by powers of 10.Convert between standard notation and scientific notation.

Objectives

The table shows relationships between several powers of 10.

Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left.

The table shows relationships between several powers of 10.

Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right.

Find the value of each power of 10.

Example 1: Evaluating Powers of 10

Start with 1 and move the decimal point six places to the left.

A. 10–6 C. 109 B. 104

1,000,000,000

Start with 1 and move the decimal point four places to the right.

Start with 1 and move the decimal point nine places to the right.

10,0000.000001

You may need to add zeros to the right or left of a number in order to move the decimal point in that direction.

Writing Math

Check It Out! Example 1

Find the value of each power of 10.

a. 10–2 c. 1010 b. 105

10,000,000,000100,0000.01

Start with 1 and move the decimal point two places to the left.

Start with 1 and move the decimal point five places to the right.

Start with 1 and move the decimal point ten places to the right.

If you do not see a decimal point in a number, it is understood to be at the end of the number.

Reading Math

Write each number as a power of 10.

Example 2: Writing Powers of 10

A. 1,000,000

The decimal point is six places to the right of 1, so the exponent is 6.

B. 0.0001 C. 1,000

The decimal point is four places to the left of 1, so the exponent is –4.

The decimal point is three places to the right of 1, so the exponent is 3.

Check It Out! Example 2

Write each number as a power of 10.

a. 100,000,000 b. 0.0001 c. 0.1

The decimal point is eight places to the right of 1, so the exponent is 8.

The decimal point is four places to the left of 1, so the exponent is –4.

The decimal point is one place to the left of 1, so the exponent is –1.

You can also move the decimal point to find the value of any number multiplied by a power of 10. You start with the number rather than starting with 1.

Multiplying by Powers of 10

Find the value of each expression.

Example 3: Multiplying by Powers of 10

A. 23.89 108

23.8 9 0 0 0 0 0 02,389,000,000

Move the decimal point 8 places to the right.

B. 467 10–3

4 6 70.467

Move the decimal point 3 places to the left.

Check It Out! Example 3

Find the value of each expression.

a. 853.4 105

853.4 0 0 0 0 Move the decimal point 5 places to the right.

85,340,000

b. 0.163 10–2

0.0 0163

0.00163

Move the decimal point 2 places to the left.

Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied.

The first part is a number that is greater than or equal to 1 and less than 10.

The second part is a power of 10.

Example 4A: Astronomy Application

Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km.

Write Saturn’s diameter in standard form.

1 2 0 0 0 0 120,000 km

Move the decimal point 5 places to the right.

Write Saturn’s distance from the Sun in scientific notation.

1,427,000,000

1,4 2 7,0 0 0,0 0 0

9 places

Example 4B: Astronomy Application

Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km.

Count the number of places you need to move the decimal point to get a number between 1 and 10.

Use that number as the exponent of 10. 1.427 109 km

Standard form refers to the usual way that numbers are written—not in scientific notation.

Reading Math

Check It Out! Example 4a

Use the information above to write Jupiter’s diameter in scientific notation.

143,000 km

1 4 3 0 0 0

5 places

Count the number of places you need to move the decimal point to get a number between 1 and 10.

Use that number as the exponent of 10. 1.43 105 km

Check It Out! Example 4b

Use the information above to write Jupiter’s orbital speed in standard form.

1 3 0 0 0 Move the decimal point 4 places to the right.

13,000 m/s

Order the list of numbers from least to greatest.

Example 5: Comparing and Ordering Numbers in Scientific Notation

Step 1 List the numbers in order by powers of 10.

Step 2 Order the numbers that have the same power of 10

Order the list of numbers from least to greatest.

Check It Out! Example 5

Step 1 List the numbers in order by powers of 10.

Step 2 Order the numbers that have the same power of 10

2 10-12, 4 10-3, 5.2 10-3, 3 1014, 4.5 1014, 4.5 1030

Lesson Quiz: Part IFind the value of each expression.

1.

2.

3. The Pacific Ocean has an area of about 6.4 х 107

square miles. Its volume is about 170,000,000

cubic miles.

a. Write the area of the Pacific Ocean in standard

0.00293

3,745,000

form.

b. Write the volume of the Pacific Ocean in scientific notation. 1.7 108 mi3

Lesson Quiz: Part II

Find the value of each expression.

4. Order the list of numbers from least to greatest