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Chapter 5—Powers, Exponents, and Roots

5-7 Working with Scientific Notation

Recall

Scientific notation is properly expressed (normalized) when there is only one non-zero digit to the left of the decimal point in the coefficient.

 

Multiplying and Dividing with Scientific Notation

The math procedures for multiplying and dividing terms expressed in scientific notation is no different from multiplying and dividing terms in any power-0f-ten format. The only thing unique about scientific notation is that the solution is given in the normalized form.

Procedure

To multiply values expressed in scientific notation

  1. Multiply the coefficients
  2. Add the exponents
  3. Normalize for scientific notation, if necessary

Examples

Problem:(2 x 103)(4 x 102) = ______

1. Multiply coefficients and add the exponents:

(2 x 103)(4 x 102) = 2 4 x 103+2

2 4 x 103+2 = 8 x 105

2. Normalize for scientific notation

8 x 105 is normalized

Solution: (2 x 103)(4 x 102) = 8 x 105

 

Problem:  (8 x 103)(4 x 104) = ______

1. Multiply the coefficients and add the exponents

 (8 x 103) x (4 x 104) = 8 4  x 103+4 = 32 x 107

2. Normalize for scientific notation

 32 x 107 = 3.2 x 108

Solution:(8 x 103)(4 x 104) = 1.28 x 1013

 

Examples & Exercises

Multiplying with Scientific Notation

Multiply these terms and formalize the solution if necessary

 

 

Procedure

To divide values expressed in scientific notation

  1. Divide the coefficients (
  2. Subtract the exponents
  3. Normalize for scientific notation

Examples

Problem: (8 x 106) (4 x 104) = _____

1. Divide the coefficients and add the exponents:

(8 x 106) (4 x 104) = 8/4 x 106-4 = 2 x 102

2. Normalize for scientific notation:

 2 x 102

Solution: (8 x 106) (4 x 104) = 2 x 102

 

Problem: (16 x 10-4)  (0.5 x 102) = _____

1. Divide the coefficients and add the exponents:

16/0.5 x 10-4-2 = 32 x 10-6

2. Normalize for scientific notation:

32 x 10-6 = 3.2 x 10-5

Solution: (16 x 10-4)  (0.5 x 102) = 3.2 x 10-5

 

Endless Examples and Exercises

Dividing with Scientific Notation

Divide these terms and normalize the solution if necessary

 

Mixed Multiplication and Division

 

Examples

 

Problem:  Perform the operations and show the results in normalized scientific notation.

(1.2 x 102)(4.5 x 103)
3 x 104

Procedure:

1. Complete the multiplication in the numerator:

(1.2 x 102)(4.5 x 103) A =A 1.2 + 4.5 x 102+3  A =A 5.4 x 105
3 x 104 3 x 104 3 x 104

2. Complete the division:

5.4 x 105  A =A 1.8 x 105-4  = 1.8 x 101
3 x 104

The result is already in normalized form.

Solution:

(1.2 x 102)(4.5 x 103) A =A 1.8
3 x 104

 

 

Endless Examples and Exercises

Mixed Multiplication and Division

Complete these  operations, presenting the solution in normalized scientific notation rounded to two decimal places.

 

Adding and Subtracting with Scientific Notation

The rules for adding and subtracting values in scientific notation are perhaps slightly more complicated than multiplication and division -- addition and subtraction requires that the exponents for the base are the same.

These terms can be added, because their exponents are equal:

The following terms can also be added,

but only after adjusting to make the exponents equal:

Notice that (3.45 x 105) was rewritten as (3450 x 102)
but the solution would have been the same by rewriting
(12.6 x 102) as (0.0126 x 105)

 

Procedure

To add or subtract values expressed in scientific notation

  1. Adjust to produce identical exponents,  if necessary
  2. Add or subtract the coefficients, as designated
  3. Attach the common power of ten
  4. Normalize for scientific notation, if necessary

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015