scientific notation scientific notation is a compact way of writing very large or very small numbers

Scientific Notation

• Scientific Notation is a compact way of writing very large or very small numbers.

A number is expressed in scientific notation when it is written as the product of a factor and a power of 10.

The factor must be greater than or equal to 1 and less than 10.

51032.8 Power of 10Factor

A number is expressed in standard form when it is the product of a factor and a power of 10.

000,8321032.8 5 Power of 10Factor

Standard Form

Scientific Notation to Standard Form

1107.8

Solve the expression to change a problem written in scientific notation to standard form.

107.8 87

Write the power in expanded form.

Simplify.

Standard Form is the answer to the expression.

Example 1:


2107.8


10107.8 870


Simplify.


Example 2:


3107.8


1010107.8 8,700


Simplify.


Example 3:


Expression Expanded Form Standard Form

87

870

8,700

1107.8 2107.8 3107.8

107.8 10107.8 1010107.8

If 8.7 is multiplied by a positive power of 10, what relationship exists between the decimal point's new position and the exponent? Why?

Multiplying by a positive power of 10 moves the decimal point to the right the number of times of the exponents value because positive powers are the result of repeated multiplication.


1107.8


10

17.8

0.87


Simplify.


Example 4:


2107.8


10

1

10

17.8

0.087


Simplify.


Example 5:


3107.8


10

1

10

1

10

17.8

0.0087


Simplify.


Example 6:


Expression Expanded Form Standard Form

.87

.087

.0087

1107.8 2107.8 3107.8

1017.8 101

1017.8

101

101

1017.8

If 8.7 is multiplied by a negative power of 10, what relationship exists between the decimal point's new position and the exponent? Why?

Multiplying by a negative power of 10 moves the decimal point to the left the number of times of the exponents value because negative powers are the result of repeated division.

Key Concept – Scientific Notation to Standard Form

• Multiplying by a positive power of 10 moves the decimal point to the right the number of times of the exponents value.

• Multiplying by a negative power of 10 moves the decimal point to the left the number of times of the exponents value.

3

Ex: Write 5.34 x 104 in standard form.

41034.5 1010101034.5

53,400

1. Write the power in expanded form.

2. Simplify.

Method 1:

ORMethod 2:

1. Determine how many places the decimal is going to move.

2. Determine which direction the decimal is going to move.

41034.5

5 40 0

53,4001 2 3 4

2

Ex: Write 3.27 x 10-3 in standard form.

31027.3

10

1

10

1

10

127.3

0.00327

1. Write the power in expanded form.

2. Simplify.

Method 1:

ORMethod 2:

1. Determine how many places the decimal is going to move.

2. Determine which direction the decimal is going to move.

3 70

0.00327123

31027.3

0

Write each number in standard form.

1. 7.42 x 105

2. 6.1 x 10-2

3. 3.714 x 102

Standard Form to Scientific NotationTo write a number in scientific notation, follow

these steps.

1. Move the decimal point to the right of the first nonzero digit.

2. Count the number of places you moved the decimal point.

3. Find the power of 10. If the original number was between 0 and 1, the exponent is negative. Otherwise, the exponent is positive.

Write 3,725,000 in scientific notation

Step 1: Move the decimal point to the right of the first nonzero digit.

3,725,000

Moving the decimal to the right of the first nonzero number finds the factor part of the scientific notation.

What is the factor? 3.725

Why do you need to move the decimal to the right of the first nonzero digit?


Step 2: Count the number of places you moved the decimal point.

3,725,000123456

Counting the number of places you moved the decimal point tells you what your exponent will be.

Why do you need to count the number of places you moved the decimal point?


Step 3: Find the power of 10. If the original number was between 0 and 1, the exponent in negative.

Otherwise, the exponent is positive.

3,725,000Is the original number between 0 and 1? No.

Is the exponent positive or negative? Positive.

What is the power of 10? 106

6


Step 4: Write 3,725,000 in scientific notation.

3.725 x 106

Remember: A number is expressed in scientific notation when it is written as the product of a factor

and a power of 10.

Factor Power of 10

Write 0.000316 in scientific notation

Step 1: Move the decimal point to the right of the first nonzero digit.

0.000316

Moving the decimal to the right of the first nonzero number finds the factor part of the scientific notation.

What is the factor? 3.16

Why do you need to move the decimal to the right of the first nonzero digit?


0.000316

Step 2: Count the number of places you moved the decimal point.

1 2 3 4

Counting the number of places you moved the decimal point tells you what your exponent will be.

Why do you need to count the number of places you moved the decimal point?


0.0003164

Step 3: Find the power of 10. If the original number was between 0 and 1, the exponent is negative.

Otherwise, the exponent is positive.

Is the original number between 0 and 1? Yes.

Is the exponent positive or negative? Negative.

What is the power of 10? 10-4

Step 4: Write 0.000316 in scientific notation.

Remember: A number is expressed in scientific notation when it is written as the product of a factor

and a power of 10.

Factor Power of 10


3.16 x 10-4

Write each number in scientific notation.

1. 14,140,000

2. 0.00876

3. 0.114

scientific notation scientific notation is a compact way of writing very large or very small numbers

Documents

factor standard form

number of times

exponents value

decimal points new position

positive powers

negative powers

result of repeated division

compact way of writing