Problem Solving with Scientific Notation .
You know that a number is in scientific notation when it is broken up as the product of
two parts. The first part, the coefficient, is a number between 1 and 10. The second
part is a power of ten. For example, 3 500 is expressed in scientific notation as
3 500 = 3.5 x 103
↑
↑
coefficient
power of ten
How can you do calculations with numbers expressed in scientific notation? First
consider addition and subtraction, then multiplication and division.
Addition and Subtraction
Like Exponents: If two numbers have like exponents, simply add or subtract the
coefficients and keep the same power of ten. Convert the sum or difference to scientific
notation if needed. For example:
a. (9.0 x 103) + (2.5 x 103) = 11.5 x 103 = 1.15 x 104
b. (4.4 x 10-5) - (2.2 x 10-5) = 2.2 x 10-5
Unlike Exponets: If two numbers have unlike exponents, they must be made the same
before the numbers can be added or subtracted. Move decimal points as needed to
compensate for changes you make to the exponents. For example:
a. (3.0 x l05 m) + (2 x 104 m) = (30 x 104 m) + (2 x 104 m) = 32 x 104 m = 3.2 x l05 m
b. (6.0 x 10-6 kg) - (4 x 10-7 kg) = 6.0 x 10-6 kg - 0.4 x 10-6 kg = 5.6 x 10-6 kg
Unlike Units: To add and subtract numbers in scientific notation with unlike units, you
need to know about metric prefixes. (Look in your text for a list of them.) To begin,
convert measurements to a common metric unit. Then make powers of ten the same.
Finally you can add or subtract. For example:
a. 6.1m + 24km = 6.1m + 2400m = 2406.1 m
b. (4.62 x 10-2 L) + (2.1 mL) = 46.2 mL + 2.1 mL = 48.3 mL = 4.83 x 101 mL
Multiplication and Division
Numbers expressed in scientific notation don't need to have the same exponents to be
multiplied or divided. Just use the following rules.
Multiplication: To multiply two or more numbers in scientific notation, multiply the
coefficients and add the exponents. Units are multiplied. For example:
a. (4 x 105 m)(2 x 106 m) = 8 x 1011 m2
b. (2 x 10-2 m)(4 x 106 m) = 8 x 104 m2
c. (3 x 103 kg)(5 x 106 m) = 15 x 109 kg. m = 1.5 x 1010 kg. m
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Division: To divide two or more numbers in scientific notation, divide the coefficients
and subtract the exponient of the denominator from the exponent of t:he numerator.
Units are divided. For example:
a. (9 x 106 m) / (3 x 102s) = 3 x 106-2m/s = 3 x 104m/s
b.(4 x 103g) / (2 x 10-2L) = 2 x 103-(-2)g/L = 2 x 105g/L
Try These:
1. (3 x 103) + (2 x 103)
2. (2 x 10-7 m) + (3 x 10-7 m)
3. (8 x 10-8 m2) – (3 x 10-8 m2)
4. (3.8 x 10-7 m2) – (2.8 x 10-7 m2)
5. (5.0 mm) + (2 x 10-4 m)
6. (6.2 km) – (3 x 102 m)
7. (2 x 105 m)(3 x 106 m)
8. (5 x 10-4 m)(4 x 10-2 m)
9. (1.50 x 10-7 m)(2.50 x 1015 m)
10. (9 x 108 kg)/(3 x l04 m2)
11. (2.4 x 105 kg)(3 x 104 m) / (4 x 10-2 s2)
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