NOTE There is often more than one way to sim-
plify expressions involving negative exponents.
In Example 8(e),we began by using the quo-
tient rule. At the right, we simplify the same
expression by using one of the special rules for
exponents. The final result is the same.
SECTION 5.1 Integer Exponents and Scientific Notation 271
Example 8(e)
Use
= Product rule
y^7
x^6
a-n
b-m=
bm
= an.
y^5 y^2
x^4 x^2
x-^4 y^2
x^2 y-^5
OBJECTIVE 6 Use the rules for exponents with scientific notation. The
number of one-celled organisms that will sustain a whale for a few hours is
400,000,000,000,000, and the shortest wavelength of visible light is approximately
0.0000004 m. It is often simpler to write such numbers in scientific notation.
In scientific notation, a number is written with the decimal point after the first
nonzero digit and multiplied by a power of 10.
Scientific Notation
A number is written in scientific notationwhen it is expressed in the form
a: 10 n, where and 1 ◊ a< 10 nis an integer.
8500 = 8.5 1000 =8.5 103 In scientific notation
It is customary to use rather
than for multiplication.
*
Notin scientific notation
0.230 is less than 1. 46.5 is greater than 10.
0.230 104 46.5 10 -^3
To write a number in scientific notation, use the following steps. (If the number
is negative, ignore the negative sign, go through these steps, and then attach a nega-
tive sign to the result.)
It is helpful to remember that, forn»1, 10 n< 1 and 10 n »10.
Converting a Positive Number to Scientific Notation
Step 1 Position the decimal point.Place a caret, ^, to the right of the first
nonzero digit, where the decimal point will be placed.
Step 2 Determine the numeral for the exponent.Count the number of
digits from the decimal point to the caret. This number gives the
absolute value of the exponent on 10.
Step 3 Determine the sign for the exponent.Decide whether multiplying
by should make the result of Step 1 greater or less. The expo-
nent should be positive to make the result greater. It should be neg-
ative to make the result less.
10 n
