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Chapter 5—Powers, Exponents, and Roots
57 Working with Scientific Notation
Recall Scientific notation is properly expressed (normalized) when there is only one nonzero 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 power0ften 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 10^{3})(4 x 10^{2}) = ______
1. Multiply coefficients and add the exponents:
(2 x 10^{3})(4 x 10^{2}) = 2 • 4 x 10^{3+2}
2 • 4 x 10^{3+2} = 8 x 10^{5}
2. Normalize for scientific notation
8 x 10^{5} is normalized
Solution: (2 x 103)(4 x 102) = 8 x 10^{5}
Problem: (8 x 10^{3})(4 x 10^{4}) = ______
1. Multiply the coefficients and add the exponents
(8 x 10^{3}) x (4 x 10^{4}) = 8 • 4 x 10^{3+4} = 32 x 10^{7}
2. Normalize for scientific notation
32 x 10^{7} = 3.2 x 10^{8}
Solution:(8 x 10^{3})(4 x 10^{4}) = 1.28 x 10^{13}
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 10^{6}) ¸ (4 x 10^{4}) = _____
1. Divide the coefficients and add the exponents:
(8 x 10^{6}) ¸ (4 x 10^{4}) = 8/4 x 10^{64} = 2 x 10^{2}
2. Normalize for scientific notation:
2 x 10^{2}
Solution: (8 x 10^{6}) ¸ (4 x 10^{4}) = 2 x 10^{2}
Problem: (16 x 10^{4}) (0.5 x 10^{2}) = _____
1. Divide the coefficients and add the exponents:
16/0.5 x 10^{42} = 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 10^{2}) = 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 10^{2})(4.5 x 10^{3}) 
3 x 10^{4} 
Procedure:
1. Complete the multiplication in the numerator:
(1.2 x 10^{2})(4.5 x 10^{3})  A =A  1.2 + 4.5 x 10^{2+3}  A =A  5.4 x 10^{5} 
3 x 10^{4}  3 x 10^{4}  3 x 10^{4} 
2. Complete the division:
5.4 x 10^{5}  A =A  1.8 x 10^{54 } = 1.8 x 10^{1} 
3 x 10^{4} 
The result is already in normalized form.
Solution:
(1.2 x 10^{2})(4.5 x 10^{3})  A =A  1.8 
3 x 10^{4} 
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 10^{5}) was rewritten as (3450 x 10^{2})
but the solution would have been the same by rewriting
(12.6 x 10^{2}) as (0.0126 x 10^{5})
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
