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

Section 6.5 Objectives

Powers of 10 can be used to read and write very large and very small number, such as the masses of atomic particles.

In algebra, you use more often than to represent multiplication. For example, . Scientific Notation is the exception. You write not .

baba

310a 310a

The table shows relationships between several powers of 10.

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What do you notice about the decimal point when 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.

The exponent increases by 1 and the

decimal point moves one place to the right.

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What do you notice about the decimal point when you multiply by 10?

Find the value of each power 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

Example 1: Evaluating Powers of 10

Write each number as a power 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.

Example 2: Writing Powers of 10

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

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

Reading Math

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.

A. 23.89 108

23.8 9 0 0 0 0 0 0

2,389,000,000

Move the decimal point 8 places to the right.

B. 467 10–3

4 6 7

0.467

Move the decimal point 3 places to the left.

C. 0.163 10-2

0.163

Move the decimal point 2 places to the left.

0.00163

Example 3: Multiplying by Powers of 10

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

The second part is a power of 10.

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

The second part is a power of 10.

bb aa 10or 10

Convert the following to scientific notation, if necessary.

A. 15 103

Move the decimal point 1 place to the left. Add 1 to the exponent.

1 5 103+1

B. 985 106

Example 4: Converting to Scientific Notation

1.5 104

Move the decimal point 2 places to the left. Add 2 to the exponent.

9 8 5 106+2

9.85 108

C. 0.008 10-9

.0 0 8 10-9-3

8 10-12

Move the decimal point 3 places to the right. Minus 3 to the exponent.

Example 5: 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 6: 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