top

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 Multiply the coefficients Add the exponents 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 Divide the coefficients ( Subtract the exponents 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 Adjust to produce identical exponents,  if necessary Add or subtract the coefficients, as designated Attach the common power of ten Normalize for scientific notation, if necessary