Transcript Section 8 – 1 Zero and Negative Exponents

Section 8 – 1 Zero and Negative Exponents
• Any nonzero number raised to the zero power
equals 1
a0 = 1
• For every nonzero number a and integer n,
a
n
1
 n
a
• Notice that a negative exponent does NOT
make anything negative, it moves the base
and the exponent to the other part of the
fraction
• Ex1. Write as a simple fraction 5-3
• Ex2. Write as a simple fraction with no
negative exponents 5a 2b3c 1
• Ex3. Write with no negative exponents 1
x 4
• Ex4. Evaluate 5x2y-3z4 when x = 2, y = 6,
and z = -3 (write answer as a simple
fraction)
Section 8 – 2 Scientific Notation
• Scientific notation is used to write large and
small numbers in a way that is easier to read
• A number in scientific notation is written as
the product of two factors in the form a x 10n,
where n is an integer and 1< a < 10
• It is ok to use x for multiplication when writing
numbers in scientific notation
• The exponent indicates how many places the
decimal point was moved and in which
direction
•
Ex1. Are the following numbers in scientific
notation? If no, why not?
A) 54.2 x 10-6
B) 4.32 x 108
C) 9.296 x 1023
D) .045 x 1011
•
Ex2. Write each of the following numbers
in scientific notation
A) 45,600,000,000
B) .00000253
C) 459.2 x 1012
• The way we typically write numbers is
standard notation
• Ex3. Write 5.82 x 108 in standard notation
Section 8-3 Multiplication Properties
of Exponents
• You can multiply numbers that have the
same base by adding their exponents:
am · an = am+n
• Simplify. Write without negative exponents
• Ex1. x3 · x5 · x · x-2
• Ex2. 4a4 · 5a-3 · a6
• Ex3. 2g-3 · 4h6 · g-2 · h-4
• To multiply numbers in scientific notation:
1) multiply the coefficients (the a)
2) multiply the powers of 10 by adding their
exponents
3) convert to scientific notation
• Ex4. Simplify. Write the answer in scientific
notation. (5 x 106)(7 x 108)
Section 8-4 More Multiplication
Properties of Exponents
• Raising a power to a power: For every
nonzero number a and integers m and n,
(am)n = amn
• Simplify. Write without negative exponents.
• Ex1. (x5)3
Ex2. a3(a5)-4
• Raising a product to a power: For every
nonzero number a and b and integer n
(ab)n = anbn
• Notice with a product to a power, everything
in the parentheses is raised to the
exponent, whether it is a number or variable
• Simplify. Write without negative exponents.
• Ex3. (5x3y2)4
• Ex4. (3a2)-4(2a3b4)2
• Ex5. m-6(3m5)2
• Ex6. (4 x 105)3
Section 8 – 5 Division Properties of
Exponents
• When you divide powers with the same
bases, subtract their exponents
am
mn
a
n
a
• Simplify. Write without negative exponents
5
12
x
• Ex1. x
Ex2.
Ex3. a 2b3
12
5
x
x
a 4b5
• Ex4. (6.8 x 109) ÷ (4 x 106)
• Raising a quotient to a power: For every
n
nonzero a and b and integer n,  a   a n
 
b
bn
• Just like with products, everything in the
parentheses is raised to the power
• Simplify. Write without negative exponents
4
3
• Ex5.  3 
Ex6.  3 
 
4
 
8
• Ex7.  5x 
 3
y 
4
Ex8.  2ab 
a3
2


5
Section 8 – 6 Geometric Sequences
• A geometric sequence is one in which you
can find consecutive numbers by multiplying
by a common ratio
• Remember that with arithmetic sequences
you were adding a common difference, now
you will be multiplying by a common number
(called the common ratio)
• If you cannot easily identify the common
ratio, divide the 2nd number by the 1st and
that is your common ratio
• To test that it is a geometric sequence and
that the ratio is constant, also divide the 3rd
number by the 2nd (you should get the
same number)
• Find the common ratio
• Ex1. 6, -18, 54, -162, …
• Ex2. 64, 48, 36, 27, …
• Ex3. Find the next two terms 8, 56, 392,
2744, …
• The formula for a geometric sequence is
A(n) = a · rn–1
• A(n) = nth term
a = first term
• r = common ratio
n = term #
• Ex4. Find the 5th and 12th terms
A(n) = 4 · (-2)n–1
• Ex5. Determine whether the sequence is
arithmetic or geometric
A) 40, 20, 10, 5, …
B) 40, 20, 0, -20, …
• Read example 5 on page 426
Section 8 – 7 Exponential Functions
• Exponential functions are in the form y = a·bx,
where a is a nonzero constant, b is greater
than 0 and b ≠ 1, and x is a real number
• The graphs will be curves
• Ex1. Evaluate y = 5 · 2x for x = -2, 2, 4
• Open your book to page 431 and look at
Objective 2 Graphing Exponential Functions
• When graphing, it is easiest to use a graphing
calculator to create these tables for you
• Ex2. Suppose 2 mice live in a barn. If the
number of mice quadruples every 3 months,
how many mice will be in the barn after 2
years?
• Ex3. Graph.
y = 2 · 3x
Section 8 – 8 Exponential Growth
and Decay
• Exponential growth is y = a·bx with a > 0 and
b >1 (a is still the starting amount and b is
called the growth factor)
• The growth factor must be greater than 1 for
it to be exponential growth (this means that
the amount is increasing)
• Compound interest is a type of exponential
growth (a = initial deposit, b = 100% +
interest rate, x = # of interest periods)
• Exponential decay uses the same model as
exponential growth, except the growth factor
b is between 0 and 1 (so it is called the
decay factor)
• The amount is decreasing in exponential
decay (100% – rate of decrease)
• Ex1. a) Suppose you deposit $1000 in a
college fund that pays 7.2% interest
compounded annually. Find the account
balance after 5 years
b) find the account balance if interest is paid
quarterly instead of yearly
• Ex2. Suppose the population of a certain
endangered species has decreased 2.4%
each year. Suppose there were 60 of these
animals in a given area in 1999. a) Write
an equation to model the number of animals
in this species that remain alive in that area.
b) Use your equation to find the
approximate number of animals remaining
in 2005.