PERFORMING CALCULATIONS IN SCIENTIFIC

NOTATION

ADDITION AND SUBTRACTION

Review: From our like term investigation:

Values we could add/subtract without adjustment

Values we could NOT add/subtract without adjustment

4 x 106

+ 3 x 106

IF the exponents are the same, we simply add or subtract the coefficient and bring the base of ten with the exponent down unchanged.

7 x 106

_______________

Adding/Subtracting SN

4 x 106

+ 3 x 105

If the exponents are NOT the same, we must move a decimalto make them the same.

It doesn’t matter what exponent you change, but changing the smaller one keeps your final

answer in scientific notation

Adding/Subtracting SN

Determine which of the numbers has the smaller exponent.

1. Change this number by moving the decimal place to the

left and raising the exponent, until the exponents of both

numbers agree.

Note: that this will take the lesser number out of scientific

notation.

4.00 x 106

+ 3.00 x 105+ .30 x 106

Move the decimal on the smallernumber to the left and raise the exponent !

4.00 x 106

Note: This will take the lesser number out of scientific notation.

Now that both numbers have common exponents

with a base of 10, they are like terms.

This allows them to be added/subtracted easily.

2. Add or subtract the coefficients as needed to get

the new coefficient.

3. The answer’s exponent will be the exponent that

both numbers have in common.

4.00 x 106

+ 3.00 x 105+ .30 x 106

4.30 x 106

Add or subtract the coefficients as needed to get the new coefficient.

The exponent will be the exponent that both numbers share.

4.00 x 106

Make sure your final answer isin the specified form, either

scientific notation or standard form.

If it is not, convert it to the correct form!

A Problem for you…

2.37 x 10-6

+ 3.48 x 10-4

2.37 x 10-6

+ 3.48 x 10-4

Solution…2.37 x 10-6

+ 3.48 x 10-4

Solution…0.0237 x 10-4

3.5037 x 10-4

PERFORMING CALCULATIONS IN SCIENTIFIC

NOTATION

MULTIPLYING AND DIVIDING

Exponent Review: Simplify each expression.

• The rule for simplifying exponents when multiplying two expressions is ______________ the exponents.

• The rule for simplifying exponents when dividing two expressions is ______________ the exponents.

• For the coefficients, or numbers in front of the variables, you ___________ or _____________ like normal.

add

subtract

multiply divide

Numbers in scientific notation are expressions too! Therefore, we’re going to use all the rules we already know to

complete operations on numbers in scientific notation.

When multiplying with scientific notation:

1.Multiply the coefficients together.

2.Add the exponents because they have the same base.

3.The base will remain 10.

4.Make sure you answer is in correct scientific notation.

(2 x 103) • (3 x 105) =

6 x 108

((9.2 x 105)(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 x 10-8

(4.6x108) (5.8x106) =26.68x1014

Notice: What is wrong with this example?

Although the answer is correct, the number is not in scientific notation.

To finish the problem, move the decimal one

space left and increase the exponent by

one.

26.68x1014 = 2.668x1015

When dividing with scientific notation

1.Divide the coefficients

2.Subtract the exponents because they have the same base.

3.The base will remain 10.

4.Make sure you answer is in correct scientific notation.

(8 • 106) ÷ (2 • 103) =

4 x 103

(1.6 x 1014)(4 x 108)

.4 x 106

4 x 105

Please multiply the following numbers.

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

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

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

4. (4.55 x 10-14) x (3.77 x 1011) =

5. (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 Standard

Notation to Scientific NotationEx. 6800

6800 1. Move decimal to get

a single digit # and

count places moved

2. Answer is a single

digit number times

the power of ten of

places moved.

68 x 10 3

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 is

expressed as a decimal number between 1 and 10

multiplied by a power of 10 ( eg, 7000 = 7 x 103 or

0.0000019 = 1.9 x 10 -6)

It’s a shorthand way of writing very large or very

small numbers used in science and math and

anywhere we have to work with very large or very

small numbers.

Why do we use it?

Changing from Scientific

Notation to Standard NotationEx. 4.5 x 10-3

1. Move decimal the same

number of places as the

exponent of 10.

(Right if Pos. Left if Neg.)

00045123

Multiply two numbers

in Scientific Notation(3 x 104)(7 x 10–5)

1. Put #’s in ( )’s Put

base 10’s in ( ) ’s

2. Multiply numbers

3. Add exponents of 10.

4. Move decimal to put

Answer in Scientific

Notation

= (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 Scientific

Notation

6.20

8.0

10-5

103

9.54x107 miles

1.86x107 miles

per second

Addition and subtraction

Scientific Notation

1. Make exponents of 10 the same

2. 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 10 7

= 1.937 x 107

1. Make exponents of 10 the same

2. Subtract 2.0 - .063 and

keep the 107 intact

Scientific

Notation

Makes

These

Numbers

Easy