Section 4.9: Scientific Notation

Patterns in Scientific Notation5 x 104 = 5 x 10,000 = 50,0005 x __ = 5 x 1,000 = 5,0005 x 102 = 5 x ___ = 5005 x 101 = 5 x 10 = 505 x 100 = 5 x 1 = 55 x 10__ = 5 x .1 = 0.55 x 10-2 = 5 x ___ = 0.05_ x 10-3 = 5 x .001 = 0.0055 x 10-4 = 5 x .0001 = 0.0005

Did You See It?

5 x 104 = 5 x 10,000 = 50,000

5 x 103 = 5 x 1,000 = 5,000

5 x 102 = 5 x 100 = 500

5 x 101 = 5 x 10 = 50

5 x 100 = 5 x 1 = 5

5 x 10-1 = 5 x .1 = 0.5

5 x 10-2 = 5 x .01 = 0.05

5 x 10-3 = 5 x .001 = 0.005

5 x 10-4 = 5 x .0001 = 0.0005

Scientific Notation• Is a shorthand way of writing numbers using

powers of 10. (Exponents!)• Scientific notation lets you know the size of a

number without having to count digits.• Write a number in scientific notation as the

product of two factors.• 7,500,000,000,000 = 7.5 x 1012

• The exponent is the number of times the decimal is moved so that it lies between the new ones and tenths place and the number to the left of the decimal is between 1 and 10.

Scientific Notation

• 10 to the 3rd power means the numbers in the thousands.

• 10 to the 6th power means the numbers is in the millions.

• 10 to the 9th power means the number is in the billions.

Visitors to the Statue of Liberty

About 4,200,000 people visit the Statue of Liberty every year. Write this number in scientific notation.

Move the decimal point to get a decimal greater than 1 but less than 10.

4,200,000 4.200000Drop the zeros after the 2. 4.2The decimal point removes 6 places to the left.Use 6 as the exponent of 10.There are 4.2 x 106 visitors every year.

Convert to scientific notation.

• 54,500,000• 723,000• 602,000,000,000• 0.00021• 0.00000005• 0.0000000000803

From Scientific to Standard Notation• You can change expressions from

scientific notation to standard notation by simplifying the product of the two factors.

• 8.9 x 105 = • Add zeros while moving the decimal

point.• Rewrite in standard notation.• 890,000

Write in Standard Notation

• 2.71 x 10-6 = • 3.21 x 107 =• 5.9 x 10-8 =• 1.006 x 1010 =• Hint: negative exponents make numbers

very small, where as positive exponents make numbers very large.

Comparing and Ordering

First, compare the powers of 10.Then, compare the decimals.

If they are NOT in correct Scientific Notation in the beginning, then convert them into CORRECT scientific notation BEFORE comparing and ordering.

Compare and Order These:

0.064 x 108 312 x 102 0.58 x 107

• Write each in scientific notation.

6.4 x 106 3.12 x 104 5.8 x 106

• Order the powers of 10. Arrange the decimals with the same power of 10 in order.

3.12 x 104 5.8 x 106 6.4 x 106

• Write the original numbers in order.

312 x 102 .58 x 107 0.064 x 108

Calculating with Scientific Notation

You can multiply numbers in scientific notation using the Rule for Multiplying Powers with the Same Base.

In this case, our SAME BASE, is 10.

To multiply exponents with the same base, then just add the exponents.

Example: 107 x 102 = 107+2=9 = 109

Multiplying Scientific Notation

3 x 10-7 and 9 x 103 (Multiply and express result in scientific notation)

(3 x 10-7)(9 x 103) = 3 x 9 x 10-7 x 103

27 x 10-7 x 103

27 x 10-4

2.7 x 101 x 10-4

2.7 x 10-3

Multiply:

• (4 x 104)(6 x 106)

• (7.1 x 10-8)(8 x 104)