Transcript Ch.7.notes_x - Windsor C

Algebra I
Chapter 7 Notes
Rules of Exponents
Section 7-1
Monomial –
Constant –
Base –
Exponent -
Section 7-1
Monomial – a number, a variable, or the product of
a number and variable with non-negative, integer
exponents
Constant – a monomial that is a real number
Base – term being multiplied in an exponential
expression
Exponent – the number of times the base is
multiplied in an exponential expression
Section 7-1: Multiplication Rules of
Exponents, Day 1
Ex) Determine whether each expression is a
monomial. Write yes or no, explain WHY.
a) 10 b) f + 24 c) x -5 d) h 2 e) -5y
Section 7-1: Multiplication Rules of
Exponents, Day 1
Product of Powers
Words
To multiply two powers that have the same base, you add their
exponents
Symbols
For any real number a, and any integers m and p:
Examples
x3 × x 5 =
am × a p = a m+p
2 ×2 =
Ex) Simplify each expression
a) (6n3 )(2n 7 )
b) (3pt 3 )(p3t 4 )
7
10
c) (-4rx 2t 3 )(-6r 5 x 2t)
Section 7-1: Multiplication Rules of
Exponents, Day 1
Power of a Power
Words
To raise a power to another power, you multiply the exponents
Symbols
For any real number a, and any integers m and p,
Example
(x 3 )5 =
Ex) Simplify
a) [(23 )2 ]4
(2 7 )10 =
b) [(2 2 )2 ]4
(a m ) p = a m×p
Section 7-1: Multiplication Rules of
Exponents, Day 2
Power of a Product
Words
Exponents can be distributed when terms are being multiplied
Symbols
For any real numbers a and b, and any integer m,
Examples
(-2xy3 )5 =
(ab)m = a m bm
Ex) Simplify each expression
a) (xy 4 )6 b) (4a 4b9c)2
c) (-2 f 2 g3h2 )3 d) (-3p5t 6 )4
Section 7-1: Multiplication Rules of
Exponents, Day 2
Ex) Use all rules to simplify
a) (3xy 4 )2 [(-2y)2 ]3 b) (-7ab4c)3[(2a2c)2 ]3
c) (5x 2 y)2 (2xy3z)3 (4yxz) d) (-2g3h)(-3gj 4 )2 (-ghj)2
Section 7-2: Division Rules of
Exponents, Day 1
Quotient of Powers
Words
To divide two powers with the same base, subtract the exponents
Symbols
For any nonzero number a, and any integers m and p,
Examples
x11
=
8
x
Ex. Simplify
g 3h 5
a) gh 2 =
am
= a m-p
p
a
520
=
7
5
b)
x3y4
x2 y
c)
k 7 m10 p
=
5 3
km p
Section 7-2: Division Rules of
Exponents, Day 1
Power of a Quotient
Words
To find the power of a quotient, find the power of the numerator and
the power of the denominator
m
a
a
m
For any real numbers a and b does not = zero, ( ) =
and any integer m,
b
bm
Symbols
r
( )5 =
t
3
( )4 =
5
Examples
Ex) Simplify
3
3p
a) ( )2 =
b)
7
3x 4 3
(
) =
4
c)
2y 2 2
( 3) =
3z
3
4x
d) ( 4 )3 =
5y
Section 7-2: Division Rules of
Exponents, Day 1
Simplify using division rules of exponents
4 2
12 3
m
p
p
1)
2) t r
3) c 4 d 4 f 3
4)
2
m p
2
p tr
c2 d 4 f 3
3xy 4 2
( 2 )
5z
2r 3t 6 4
(
)
3
4rt
3m 5r 3 3
(
)
2
6m r
5)
6)
Section 7-2: Division Rules of
Exponents, Day 2
Zero Exponent Property
Words
Any nonzero number raised to the zero power is equal to 1
Symbols
For any nonzero number a,
an0p =1
-5
Example
4
b r0 =
( ) =
c
15 =
-2
0
Ex) Simplify. Assume no denominator = zero
2 5 2
4 2 0
5 0
4n
q
r
b
0
a) (- 3 2 ) b) x y = c) c d =
9n q r
x
3
b2c
Section 7-2: Division Rules of
Exponents, Day 2
Negative Exponent Property
Words
For any nonzero number a, and any integer n,
Symbols
For any nonzero number a and integer n,
Examples
1
1
2 = 4=
2 16
-4
or
1
=
-4
j
a
a-n =
3m 5r 3 3
)
46m 2 r
j
(
-nis the reciprocal of
1
an
Ex) Simplify. NO NEGATIVE EXPONENTS!
-5 4
-3 4
2 3 -5
-3
2
n
p
5r
t
2a
b
c
v
wx
a) -2 = b)
= c)
= d)
=
2 7 -5
-3 -1 -4
r
-20r t u
10a b c
wy-6
1
an
Section 7-2: Division Rules of
Exponents, Day 2
Simplify using division rules of exponents
1) (4k m )
2) 20qr t
3
2 3
(5k 2 m-3 )-2
3)
-3x -6 y -1z -2 -2
(
)
-2
-5
6x yz
-2 -5
4q 0 r 4t -2
-2 4 2
2a
b c -1
4) (
)
-2 -5 -7
-4a b c
Section 7-4: Scientific Notation
Scientific Notation – a number written in the
form a ´10n, where 1 < a < 10 and n is an integer.
Ex) Write the following numbers in scientific
notation.
1) 201,000,000
2) 0.000051
Section 7-4: Scientific Notation
Ex) Write the following numbers in standard
form
1) 6.3´10 9
2) 4 ´10-7
Section 7-4: Multiplying with Scientific
Notation
Ex) Use rules of exponents to multiply the
following numbers together. Express your
answer in both scientific notation and standard
form!
1) (3.5´10-3 )(7´105 )
2) (7.8´10-4 )2
Section 7-4: Dividing with Scientific
Notation
Ex) Use rules of exponents to multiply the
following numbers together. Express your
answer in both scientific notation and standard
form!
1) 3.066 ´10
2) 1.305´10
8
7.3´10 3
3
1.45´10-4
Section 7-5: Exponential Functions –
Exponential Growth, Day 1
Exponential Function – A function that can be
x
written in the form y = ab
, where
a cannot be 0, b >
0, and b cannot be 1. Examples of exponential
1 x
x
x
functions: y = 2(3) , y = 4 , or y = ( 2 )
Exponential Growth
Equation
Domain and
Range
Intercepts
End Behavior
f (x) = ab x where a > 0, b >1
Section 7-5: Exponential Functions –
Exponential Growth, Day 1
Graph of Exponential Growth
Section 7-5: Exponential Functions –
Exponential Growth, Day 1
Ex) Graph y = 3x, Find the y-intercept, and state
the domain and range. You will have to create a
table to graph! What is the pattern on the table?
x
-2
-1
0
1
2
3x
y
Section 7-5: Exponential Functions –
Exponential Decay, Day 2
Exponential Decay
Equation
Domain and
Range
Intercepts
End Behavior
f (x) = ab x
where
a > 0, 0 < b <1
Section 7-5: Exponential Functions –
Exponential Decay, Day 2
Graph of Exponential Decay
Section 7-5: Exponential Functions –
Exponential Decay, Day 2
Ex) Graph y = (13) , Find the y-intercept, and state
the domain and range. You will have to create a
table to graph! What is the pattern on the table?
x
x
-2
-1
0
1
2
1
( )x
3
y
Section 7-6: Exponential Growth and
Decay Patterns, Day 1
Equation for Exponential Growth: y = a(1+ r)t
a: initial amount t : time
y: final amount
r: rate of change expressed as a decimal, r > 0
Ex) The prize for a radio station contest begins with a $100 gift card. Once a day, a
name is announced. The person has 15 minutes to call or the prize increases 2.5%
for the next day.
a) Write an equation representing the amount of the gift card after t days with
no winner
b) How much will the card be worth if no one claims it after 10 days?
Section 7-6: Exponential Growth and
Decay Patterns, Day 1
Compound Interest – interest earned or paid on both the initial investment
AND previously earned interest. It is an application of exponential growth.
Equation for Compound Interest
r nt
A = P(1+ )
n
A: current amount
P: principal/initial amount
r: annual interest rate expressed as a decimal
n: number of times interest is compounded per year
t: time in years
Section 7-6: Exponential Growth and
Decay Patterns, Day 1
Ex) Maria’s parents invested $14,000 at 6% per
year compounded monthly. How much money
will there be in the account after 10 years?
Section 7-6: Exponential Growth and
Decay Patterns, Day 2
Equation for Exponential Decay
y = a(1- r)
t
a: initial amount y = final amount
t: time
r: rate of decay as a decimal 0 < r < 1
Ex) A fully inflated raft is losing 6.6% of its air every day. The raft originally
contained 4500 cubic inches of air.
a) Write an equation representing the loss of air
b) Estimate the amount of air in the raft after 7 days
Section 7-6: Exponential Growth and
Decay Patterns, Day 2
Solve the 3 problems. You must choose which equation to use on each.
1) Paul invested $400 into an account with 5.5% interest compounded monthly.
How much will he have in 8 years?
1)
Ms. Acosta received a job as a teacher with a starting salary of $34,000. She
will get a 1.5% increase in her salary each year. How much will she earn in 7
years?
1)
In 2000 the 2200 students attended East High School. The enrollment has
been declining 2% annually. If this continues, how many students will be
enrolled in the year 2015?