Math 152 — Rodriguez
Scientific Notation
Blitzer — 1.7
I. Scientific Notation
A. Scientific notation is used to write very large and very small numbers.
B. A number is said to be written in scientific notation if it is of the form
b x 10n , where n is an integer and 1 ≤ |b| < 10
C. Scientific notation to decimal notation
If the exponent is positive, move the decimal point to the right n places.
If the exponent is negative, move the decimal point to the left |n| places.
Examples:
1) 3.1 x 104 = 3.1 x 10,000 = 31,000
Move the decimal point to the right 4 places.
2) 4.5 x 10− 3 = 4.5 x 0.001 = 0.0045
Move the decimal point to the left 3 places.
3) −2.5 x 105 =
Move the decimal point ____________ places.
4) 5.25 x 10− 5 =
Move the decimal point ____________ places.
5) 9.4 x 10− 6 =
6) 6 x 105 =
D. Decimal notation to scientific notation
Steps:
(Note: my explanation is DIFFERENT than the books)
1. Determine the numerical factor, b, by looking at the number and placing the decimal
point so that 1 ≤ |b| < 10.
2. Determine the exponent n by counting how many places the decimal point was moved
to get from the numerical factor b to the original decimal number.
• Make the exponent positive if you have to move the decimal point in the numerical
factor b to the right to get the decimal number.
• Make the exponent negative if you have to move the decimal point in the
numerical factor b to the left to get the decimal number.
Examples:
1) 32,000 =
b = 3.2;
to get from 3.2 to 32,000 had to move decimal point to right 4 places so exponent is positive 4
2) −125,000=
b = −1.25 since 1 ≤ |−1.25| < 10
to get from −1.25 to −125,000 had to move decimal point to the _____________ places so
exponent is ___________; doesn’t matter that b value is negative
3) −0.00000000038 =
b=
to get from _____ to −0.00000000038 had to move decimal point to the __________ places
so exponent is _____________
4) 0.00000087 =
5) 425,000,000 =
II. Operations with Numbers in Scientific Notation
A. To multiply two numbers written in scientific notation, multiply the numerical factors and
multiply the powers of ten. Make sure the answer is written in scientific notation.
Examples:
2 × 10 3 3 × 10 4 = ( 2 ⋅ 3) × 10 3 ⋅10 4 =
1)
Done?
(
)(
)
(
)
2)
( 6 × 10 ) ( 5.5 × 10 ) = ( 6 ⋅ 5.5 ) × (10
3)
( 2.8 × 10 ) ( 4 × 10 )
−4
12
6
−14
12
)
⋅10 −4 =
Done?
4)
(1.5 × 10 ) ( 6 × 10 )
−3
−4
B. To divide two numbers written in scientific notation, divide the numerical factors and
divide the powers of ten. Make sure the answer is written in scientific notation. If
necessary, round the numerical factor in your answer to two decimal places.
Examples:
1)
2.5 × 10 −9 2.5 10 −9
=
×
=
5 10 −3
5 × 10 −3
2)
1.6 × 10 −8
8 × 10 5
Done?
3)
9.6 × 10 4
3.2 × 10 −2
III. Applications involving numbers in scientific notation
Given the information in the application, decide if you are multiplying or dividing the numbers
in the problem. (These were the only two operations we learned.) Make sure to give the final
answer in the format described (scientific, decimal, rounded to a given place value).
Example: The area of Alaska is approximately 3.66 x 108 acres. The state was purchased in
1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the U.S. pay
Russia?
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Example: The mass of one hydrogen atom is 1.67 x 10− 24 gram. Find the mass of 20,000
hydrogen atoms. Express the answer in scientific notation.
Example: Twenty billion hot dogs are consumed by Americans each year. If the consumption
of hot dogs was divided evenly among all Americans, how many hot dogs would each
American consume in a year? Use 300 million as the U.S. population. Round answer to the
nearest whole number.
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