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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 - Multiply the coefficients
- Add the exponents
- Normalize for scientific notation, if necessary
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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
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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
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