Download Negative Exponents and Scientific Notation

Goals:
Negative Exponents
and Scientific Notation
1) Simplify rational numbers
raised to negative exponents.
2) Express numbers in scientific
notation.
Review: The exponent rules:
1 as an exponent
a1 = a
0 as an exponent
a0 = 1
Product rule
Power rule
am . an = am + n
am
= a m −n
an
(am)n = amn
Product to a power
(ab)n = anbn
Quotient to a power
an
a
  = n
b
b
Quotient rule
In the quotient rule, what
happens when the exponent
in the numerator is greater
than that in the
denominator?
n
so
Example:
3
a
= a 3− 5 = a − 2
5
a
and
3
a
a⋅a⋅a
1
=
= 2
5
a
a⋅a⋅a⋅a⋅a a
a −2 =
1
a2
and in general:
Negative Exponents:
For any real number a that is
nonzero and any integer n:
a −n =
1
an
1
Examples: Evaluate.
1) 4–1
2) 2–2
Examples: Simplify by
rewriting with only positive
exponents.
1
1)
x −15
2)
xy −2
Example: The sun is 93,000,000
miles from earth.
93,000,000 = 9.3 x 10,000,000 =
9.3 x 107
since
107 = 10,000,000
Simplifying expressions involving
negative exponents involves
rewriting the expression with only
positive exponents.
Trick: When you move a factor
from numerator to denominator
or vice versa, change the sign of
the factor to obtain an equivalent
expression.
Scientific Notation
Numbers with lots of zeros can
be written compactly using
scientific notation.
The correct form for the scientific
notation of a number is a number
with only one digit before the
decimal place times 10 raised to a
power.
Correct:
9.3 x 107
Incorrect: 93 x 106
2
Example: The wavelength of the
yellow light given off by a sodium
lamp is 0.000000589 meters.
0.000000589 =
When converting to scientific
notation, count the number of
times the decimal place is
moved.
93,000,000
5.89
=
100,000,00
5.89 ×10 −7
When moving the decimal to
the left, multiply by 10 raised to
positive number of decimal
moves.
To the right – 10 to negative
number of decimal moves.
Converting from scientific
notation to decimal form:
Negative exponent - move the
decimal point the absolute value
of the exponent to the left.
Positive exponent – move the
decimal point exponent number
of places to the right.
0.000000589
Examples: Convert to scientific
notation.
1) 92,800
3) 0.000576
2) 5,700,000
4) 0.00000009
Examples:
1) 3.957 x 108
2) 4.76 x 10–5
3