Section 9.3

What we are Learning:To express numbers in scientific and standard notation

To find products and quotients of numbers expressed in scientific notation

Scientific Notation:

• a x 10n

– a ≥1 but < 10• NOT a decimal!

– n is an integer (…-2, -1, 1, 2…)• We use Scientific Notation to express very

large or very small numbers

Scientific Notation for Large Numbers:

• Express the number as a product of a number greater than or equal to 1 but less than 10, and a power of 10– Place a decimal point after the first number– Count how many numbers, including zeros are after the

decimal point• This is the number that will be your power of 10

• Example:24,000,000,0002.40000000002.4 x 1010

Scientific Notation for Small Numbers:• Numbers between zero and one• The power of 10 will be negative

– Place a second decimal point after the first non-zero number

– Count the numbers between the two decimals• This will be your negative power of 10

• Example:.000065.00006.56.5 x 10-5

Examples:

• 67,852,0006.78520006.7852 x 107

• 0.0000093450.000009.3459.345 x 10-6

Scientific Notation to Standard Notation:

• For large numbers (positive powers of 10)– Count how many numbers come after the decimal– Subtract this amount from the power of 10– This total is the number of zeros to add after the last

number

Example:3.29 x 107 (There are two numbers after the decimal point)

7 - 2 = 5 (This is how many zeros to add after the last number)

32,900,000

Scientific Notation to Standard Notation:

• For small numbers (negative powers of 10)– Count how many numbers come before the decimal

• Should ALWAYS be 1

– Add this number to the negative exponent– This is how many zeros to add before the first number

• Example:5.73 x 10-5 (there is one number before the decimal point)

-5 + 1 = 4 (this is how many zeros to add before the first number)

.0000573

Examples:

• 1.2475 x 1010

10 – 4 = 612,475,000,000

• Notice:– We can also solve the

expression by 1.22475 x 10,000,000,00012,475,000,000

• 4.7592 x 10-8

-8 + 1 = 7.000000047592

• Notice:– We can also solve the

expression by4.7592 x (1/108)

4.7592 x (1/100,000,000)4.7592/100,000,000.000000047592

Using Scientific Notation to Evaluate Expressions:

• Write numbers in Scientific Notation• Use the Associative Property to group like

terms together• Use the Properties of Powers

Examples:• (392)(4,380,000)

(3.92 x 102)(4.38 x 106)(3.92 x 4.38)(102 x 106)17.1696 x 108

We need to have a number that is ≥ 1 and < 10. So we need to move the decimal one more place to the left…

1.71696 x 109

• (.00008562)(.00064)(8.562 x 10-5)(6.4 x 10-4)(8.562 x 6.4)(10-5 x 10-4)54.7968 x 10-9

We need to have a number that is ≥ 1 and <10. So we need to move the decimal one more place to the left…

5.47968 x 10-8

When We…

• Move the decimal to the left to create 10<n≥1 we add to the exponent

• Move the decimal to the right to create 10<n≥1 we subtract from the exponent

Let’s Work These Together

Write in Scientific Notation:• 95,000,000

• 0.00000039

Write in Standard Notation:• 1.879 x 10-5

• 845 x 106

Let’s Work These Together:Evaluate; Express in Scientific Notation and Standard Notation

• (6.4 x 103)(7 x 104) • 3.372 x 10-6/4.8 x 10-2

Homework:

• Page 509, 510– 19 to 39 odd