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Home , Power of 10

Powers of Ten and

Large such as the speed of light and the distance from the Earth to the Sun are often represented using an abbreviated form.

In science, we refer to the speed of light as:

3 × 108 m/s.

This is the abbreviated form of 300 000 000 metres per second.

Powers of 10 are used to estimate and reduce the size of to make them easier to work with.

A power of 10 is a power with a base of 10.

The exponent can be any 10 Repeated Power of 10 Answer 102 10 × 10 100

The exponent is 2. There are 2 zeros in the answer.

For example: • 10 × 10 × 10 = 1000, and can also be represented as 103. • 10 × 10 × 10 × 10 × 10 = 100 000, and can also be represented as 105.

Multiples of 10 can be represented as powers of 10.

Examples

A) 7000 is 7 × 1000 and can be represented as 7 × 103. B) 20 000 is 2 × 10 000 and can be represented as 2 × 104.

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 1/7

Scientific Notation

Scientific notation is a way of writing numbers in powers of 10 and is used when communicating large numbers.

Example

The distance between Earth and the Sun is approximately 150 000 000 000 m. This number is abbreviated as:

1.5 × 1011 m This number represents a scientific notation.

Numbers in scientific notation look like this example.

In general, scientific notation looks as follows.

There may be one or more exponent digits to the right of the

decimal point × 10

A number that is at least multiplied by a power of 10 1, but less than 10

Example 1.275 × 103 (original number = 1275)

Notice that in representing numbers in scientific notation:

• there is only one number in front of the decimal point • there may be several numbers to the right of the decimal point.

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 2/7

Writing Numbers in Scientific Notation Using Repeated Multiplication

Example What is 1275 in scientific notation?

127.5 × 10 = 1275 12.75 × 10 × 10 = 12.75 1.275 × 10 × 10 × 10 = 1.275

There are three 10’s being multiplied.

Therefore: 1275 = 1.275 × 103 in scientific notation

Writing Numbers in Scientific Notation by Moving the Decimal Point

This is a good mind math method. Some people call it a shortcut.

Example What is 1275 in scientific notation?

• In all whole numbers, the decimal point is to the right of the last digit—we do not place it unless it is needed.

• The decimal point is located behind 5.

1275.

• Move the decimal point to the LEFT until there is only one number to the left of the decimal point.

1275. Æ 127.5 Æ 12.75 Æ 1.275

• Next, COUNT the number of times you moved the decimal point to the left (3 times in the example). This number is your exponent in the power of 10.

Therefore: 1 2 7 5. = 1.275 × 103 in scientific notation

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 3/7

Examples

A) What is 4000 in scientific notation?

4000 4 0 0 0. = 4 × 103

Decimal moved 3 places.

B) What is 3 000 000 in scientific notation?

3 000 000 3 0 0 0 0 0 0. = 3 × 106

Decimal moved 6 places.

C) What is 345 in scientific notation?

345 3 4 5. = 3.45 × 102

Decimal moved 2 places.

D) What is 12 065 in scientific notation rounded to the nearest tenth?

12 065 1 2 0 6 5. = 1.2065 × 104 = 1.2 × 104

after rounding

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 4/7

Practice: Working with Powers of Ten

1. Do you notice a pattern between the number of zeros following the one, and the exponent? Explain this pattern to your teacher or a classmate.

2. Write each number in repeated multiplication and as a power of 10. Number Repeated Power of 10 Multiplication 1 000 103

10 000

100 000 10 × 10 × 10 × 10 × 10 1 000 000

3. Maddy was examining the population of cities in Alberta. She found the following statistics: Calgary – 876 519 Edmonton – 648 284 Grande Prairie – 35 962 Lethbridge – 68 712 Medicine Hat – 50 512 Red Deer – 68 308

Maddy created a chart in which she rounded each number and identified each population as a power of 10. Rounded Populations Populations in Cities (to the nearest 10 000) Powers of 10 Calgary 880 000 8.8 × 106 Edmonton 650 000 6.5 × 104 Grande Prairie 40 000 4 × 104 Lethbridge 70 000 7 × 104 Medicine Hat 50 000 5 × 105 Red Deer 70 000 7 × 104

Luckily, when Maddy rechecked her work before handing it in, she identified three errors. Can you find and correct her three errors? (Hint: The errors are not in the rounded numbers.)

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 5/7

4. The chart below is a result of Census 2001. It shows the population of each of Canada’s provinces and territories. Rewrite each population figure in scientific notation. Round each number to one digit to the right of the decimal point.

Province/Territory Population Scientific Notation British Columbia 3 907 738 Alberta 2 974 807 Saskatchewan 978 933 Manitoba 1 119 583 Ontario 11 410 046 Quebec 7 237 479 New Brunswick 729 498 Nova Scotia 908 007 Prince Edward Island 135 294 Newfoundland 512 930 Yukon Territory 28 674 Northwest Territories 37 360 Nunavut 26 745

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 6/7

5. Rewrite each figure below in scientific notation. Round each number to one digit to the right of the decimal point.

Number Scientific Notation

14 532 739

27 466 570

39 774 129

47 453 982

73 283 465

96 374 102

137 845 923

177 299 374

187 378 069

566 734 292

1 456 292 349

3 744 239 129

12 373 460 595

Knowledge and Employability Studio Mathematics Numbers: Whole Numbers: ©Alberta Education, Alberta, Canada (www.LearnAlberta.ca) Powers of Ten and Scientific Notation 7/7