PERFORMING PERFORMING CALCULATIONCALCULATION

S IN S IN SCIENTIFIC SCIENTIFIC NOTATIONNOTATION

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

ReviewReview::Scientific notation Scientific notation expresses a number in the expresses a number in the form:form: M x 10M x 10nn

1 1 M M 1010

n is an n is an integerinteger

4 x 104 x 1066

+ 3 x 10+ 3 x 1066

IFIF the exponents the exponents are the same, we are the same, we simply add or simply add or subtract the subtract the numbers in front numbers in front and bring the and bring the exponent down exponent down unchanged.unchanged.

77 x 10x 1066

_______________

4 x 104 x 1066

+ 3 x 10+ 3 x 1066

IFIF the exponents the exponents are the same, we are the same, we simply add or simply add or subtract the subtract the numbers in front numbers in front and bring the and bring the exponent down exponent down unchanged.unchanged.

77 x 10x 1066

_______________

4 x 104 x 1066

+ 3 x 10+ 3 x 1055

If the exponents If the exponents are NOT the are NOT the same, we must same, we must move a decimal to move a decimal to makemake them the them the same.same.

4.00 x 104.00 x 1066

+ + 3.00 x 103.00 x 1055 + + .30 x 10.30 x 1066

4.304.30 x 10x 1066

Move the Move the decimal decimal on the on the smallersmaller number!number!

4.00 x 104.00 x 1066

A Problem for A Problem for you…you…

2.37 x 102.37 x 10-6-6

+ 3.48 x 10+ 3.48 x 10-4-4

2.37 x 102.37 x 10-6-6

+ 3.48 x 10+ 3.48 x 10-4-4

Solution…Solution…002.37 x 10002.37 x 10--

66

+ 3.48 x 10+ 3.48 x 10-4-4

Solution…Solution…0.0237 x 100.0237 x 10-4-4

3.5037 x 103.5037 x 10-4-4

Addition and subtractionScientific Notation

1. Make exponents of 10 the same2. Add 0.2 + 3 and keep the 103 intact

The key to adding or subtracting numbers in Scientific Notation is to make sure the exponents are the same.

2.0 x 102 + 3.0 x 103

.2 x 103 + 3.0 x 103

= .2+3 x 103

= 3.2 x 103

2.0 x 107 - 6.3 x 105

2.0 x 107 -.063 x 107

= 2.0-.063 x 107

= 1.937 x 107

1. Make exponents of 10 the same2. Subtract 2.0 - .063 and keep the 107 intact

PERFORMING PERFORMING CALCULATIONCALCULATION

S IN S IN SCIENTIFIC SCIENTIFIC NOTATIONNOTATION

MULTIPLYING AND DIVIDINGMULTIPLYING AND DIVIDING

Rule for Multiplication

When multiplying with scientific notation:

1.Multiply the coefficients together.

2.Add the exponents.

3.The base will remain 10.

(2 x 103) • (3 x 105) =

6 x 108

(9.2 x 105) x (2.3 x 107) =

21.16 x 1012 =

2.116 x 1013

(3.2 x 10-5) x (1.5 x 10-3) =

4.8 • 10-8

Rule for Division

When dividing with scientific notation

1.Divide the coefficients

2.Subtract the exponents.

3.The base will remain 10.

6.20 x 10–5

8.0 x 103DIVIDE USING SCIENTIFIC

NOTATION

= 0.775 x 10-8

= 7.75 x 10–9

1. Divide the #’s &Divide the powers of ten(subtract the exponents)

2. Put Answer in ScientificNotation

6.20

8.0

10-5

103

(8 • 106) ÷ (2 • 103) =

4 x 103

(3.402 x 105) ÷ (6.3 x 107) =

0.54 x 10-2

Please multiply the following numbers.

(5.76 x 102) x (4.55 x 10-4) =

(3 x 105) x (7 x 104) =

(5.63 x 108) x (2 x 100) =

(4.55 x 10-14) x (3.77 x 1011) =

(8.2 x10-6) x (9.4 x 10-3) =

Please multiply the following numbers.

(5.76 x 102) x (4.55 x 10-4) =

(3 x 105) x (7 x 104) =

(5.63 x 108) x (2 x 100) =

(4.55 x 10-14) x (3.77 x 1011) =

(8.2 x10-6) x (9.4 x 10-3) =

2.62 x 10-1

2.1 x 1010

1.13 x 109

7.71 x 10-8

1.72 x 10-2

1. (5.76 x 102) / (4.55 x 10-4) =

2. (3 x 105) / (7 x 104) =

3. (5.63 x 108) / (2) =

4. (8.2 x 10-6) / (9.4 x 10-3) =

5. (4.55 x 10-14) / (3.77 x 1011) =

Please divide the following numbers.

1. (5.76 x 102) / (4.55 x 10-4) = 1.27 x 106

2. (3 x 105) / (7 x 104) = 4.3 x 100 = 4.3

3. (5.63 x 108) / (2 x 100) = 2.82 x 108

4. (8.2 x 10-6) / (9.4 x 10-3) = 8.7 x 10-4

5. (4.55 x 10-14) / (3.77 x 1011) = 1.2 x 10-25

Please divide the following numbers.

Changing from StandardNotation to Scientific NotationEx. 6800

6800 1. Move decimal to geta single digit # andcount places moved

2. Answer is a singledigit number timesthe power of ten ofplaces moved.

68 x 103

If the decimal is moved left the power is positive.

If the decimal is moved right the power is negative.

123

What is Scientific NotationA number expressed in scientific notation isexpressed as a decimal number between 1 and 10multiplied by a power of 10 ( eg, 7000 = 7 x 10 3 or0.0000019 = 1.9 x 10 -6)

It’s a shorthand way of writing very large or verysmall numbers used in science and math andanywhere we have to work with very large or verysmall numbers.

Why do we use it?

Changing from ScientificNotation to Standard NotationEx. 4.5 x 10-3

1. Move decimal the samenumber of places as theexponent of 10.(Right if Pos. Left if Neg.)

00045123

Multiply two numbersin Scientific Notation(3 x 104)(7 x 10–5)

1. Put #’s in ( )’s Putbase 10’s in ( )’s

2. Multiply numbers3. Add exponents of 10.4. Move decimal to put

Answer in ScientificNotation

= (3 x 7)(104 x 10–5)

= 21 x 10-1

= 2.1 x 100

or 2.1

6.20 x 10–5

8.0 x 103DIVIDE USING SCIENTIFIC

NOTATION

= 0.775 x 10-8

= 7.75 x 10–9

1. Divide the #’s &Divide the powers of ten(subtract the exponents)

2. Put Answer in ScientificNotation

6.20

8.0

10-5

103

9.54x107 miles

1.86x107 milesper second

Addition and subtractionScientific Notation

1. Make exponents of 10 the same2. Add 0.2 + 3 and keep the 10 3 intact

The key to adding or subtracting numbersin Scientific Notation is to make sure theexponents are the same.

2.0 x 102 + 3.0 x 103

.2 x 103 + 3.0 x 103

= .2+3 x 103

= 3.2 x 103

2.0 x 107 - 6.3 x 105

2.0 x 107 -.063 x 107

= 2.0-.063 x 107

= 1.937 x 107

1. Make exponents of 10 the same2. Subtract 2.0 - .063 and keep the 107 intact

Scientific Notation Makes These Numbers Easy