Download Division by Binomials Part 2

MIAP 20S – Notes
Unit: Algebra and # Notes
Name ________________________
Dates Taught _________________
General
Outcome
10I.A.2

10I.A.2
10I.A.2
10I.A.3



Demonstrate an understanding of irrational
numbers by representing, identifying, and
simplifying irrational numbers and ordering
irrational numbers.
Express a radical as a mixed radical
Express a mixed radical as an entire radical
Demonstrate an understanding of powers
with integral and rational exponents
Comments : ________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
1
MIAP 20S – Notes
Outcome 10I.A.3: Integral Exponents
Note: a,b and x are rational and variable basis while m and n are integral exponents.
Law:
Example:
Converting Negative Powers
an 
2
3 
1
,a  0
n
a
Product of Powers
or
1

3
2
( x3 )( x 2 ) 
(am )(an )  am n ,a  0
Quotent of Powers
(am ) /(an )  am n ,a  0
 43 
 2  
4 
(72 ) 3 
Power of a Power
(am )n  amn
Power of a Product

(3a 2bc4 )3 
(ab)m  ambm
Power of a Quotient
(a/b)m  am /bm ,b  0
Zero Exponent
4
3
  
2
(2)0 
(2x) 0 
a0 1,a  0

2
MIAP 20S – Notes
More Examples: Simplify. Your answer should contain positive exponents only.
a)
(3r 4 s 3 )( 2r 2 s 5 ) 
b)
 4m2 


2 
 2m 
4
c)
6x 5 
f)
3x 
4 2

d)
g)
7m 5 n8 
3x 4

6 x7
e)
h)
5
  
7
 10a 4b 5

15a 5b  2
Homework: Handout: Kuta Software: Properties of Exponents
3
MIAP 20S – Notes
Outcome 10I.A.3: More Integral Exponents
Examples: Simplify. Your answer should contain positive exponents only.
3
a)
c)
e)
3 5
2
( 2r ) ( 2r ) 
 4m2n3  2m7n1  4m 

 
2 2
2m n


2a
3x
b)
3x 
2 2
d)

4v w 
0
3 3
 2v 2 w10 

1 5 0 3
b c
 2a 1bc

2 3
4a b
Homework: Handout: Kuta Software: More Properties of Exponents
4
MIAP 20S – Notes
Outcome 10I.A.3: Rational Exponents
Note: a,b and x,y are rational and variable basis while m and n are integral exponents.
Law:
Product of Powers
(a )(a )  a
m
n
Example
3
5
m n
(2 )(2 ) 
,a  0
Quotent of Powers

(am ) /(an )  am n ,a  0
Power of a Power
(a )  a
m n

mn
Power of a Product
(ab)  a b
m
m m
(a/b)  a /b ,b  0
m
m
1
3
(6 ) /(62 ) 
2
3 3
(2 ) 
1 1
2 3

(27x ) 

x 2 12
( 4) 
y
Power of a Quotient
m
4
5
(5x) 0 
Zero Exponent
a0 1,a  0
(5x) 0 



Note: A power with a rational exponent can be written with the exponent in decimal or
fractional form. Eg.
5
2
3 4  3.5 .
MIAP 20S – Notes
Extra Examples:
Example:
1.75
a)
(4
d)
/4 5 / 4 ) 
1
7 3
b)
( y 0.75 / y 6/ 4 )3 
e)
(25 /16) .7 
(8x ) 

g)
Homework: Page 72, Q #
6
c)
( x )( x ) 
f)
(4 x) 
3
2


3 2
( z ) ( z )  
5
1
3
0.3
 ( xy ) 
 ( x2 y) 


2
h)
2.5

 49a 
 36b 2 


3
i)
0.5

MIAP 20S – Notes
Outcome 10I.A.2: Number Systems and Approximating Irrationals
ie.
numbers,
N = {1, 2, 3, . . . }
, are all
ie.
numbers,
W = {0, 1, 2, 3, . . . }
, are all positive integers and
ie.
,
, are whole numbers and their
I = {. . . -2, -1, 0, 1, 2, . . . }
numbers,
.
.
, are any numbers written in the form of a
a
, where a & b are
b
and b
numbers,
the form
and the
ie. R = Q U Q
.
and b  0.
, are the
a

 |, b  I , b  0 
b

ie.
, are any number that
a
, where a & b are
b
numbers,
.
number set.

be written in
Q = Set of irrational numbers
of the
number set
Reals
Rationals
Integers
Wholes
Irrationals
Naturals
Word bank:
7
cannot
irrational
rational
fraction
irrational
real
integers
natural
union
integers
opposites
whole
integers
positive
zero
integers
rational
MIAP 20S – Notes
Examples:
1. Which Number System best represents the following numbers?
a)
2
c)
e)
35
d) –5
f) 0.131313…

g)
i)
b) 0.25
25
0.123456789…
h) 0
j) ¾
2. Write each number in decimal form (round to 2 decimal places). Some may already be written in
decimal form.
a)
3
c)
e)

g)
i)
b) 1.41
45
d) –3
f) 0.171717…

16
0.5123456789…
h) 0
j) ¾

Place the above numbers on a horizontal number line (below). Clearly label the number line and use an
appropriate scale.
Homework: MPC20S, Exercise 20
8
MIAP 20S – Notes
Outcome 10I.A.2&3 Rational Exponents
A
as follows:
is the proper name for the
r
x
of a number. The parts of a radical are
where, r is the root
x is the
p is the
p
,
,
of the radicand, and
is the
1
2
3
sign.
4
5
6
Perfect
Squares
Perfect
Cubes
Perfect 4th
Perfect 5th
Example: Simplify:
a)
4
16 
Why?
b)
3
216 
c)
4
256 
d)
5
32 
e)
2
x2 
f)
4
m8 
g)
5
k 20 
Radicals may be written as numbers with
exponents.
Example: The square root of 5,
, could also be written as
.
o The 1 represents the exponent or
of 5.
o The 2 represents the second
or
root of 5.
o When there is
root index shown as part of the radical, the root index is understood to
be
.
r
9
p
x  x x
p
r
p r
MIAP 20S – Notes
Examples 1: Write the following using a fractional exponent:
a)
a5 =
b)
3
64 =
c)
8
b5 =
Examples 2: Write the following in radical form:
5
b) 8.75 =
a) 3 2 =
5
c)
5
a7
Example 3: Evaluate without using a calculator:
Hint: Always evaluate the radical first, then apply the power (Make the expression
before you make it bigger)
a)
9
1
2
=
b)
4
0.5
=
c)
2
d)
8
3
=
e)
8
 
 18 
4
3
2
=
1
2
=
Homework: Next Page of this booklet & Page76, Q # 1, 2
10
MIAP 20S – Notes
Homework:
1.Write as a rational exponent:
a)
3
x=
1
d)
52
7
=
b)
5
32 =
c)
6
e)
3
13x 2 =
f)
3
17
94
=
x6 =
2.Write as a radical:
1
1
a)
n4 =
d)
45 =
0.6
8
b)
11 2 =
c)
e)
5b  =
1
3 5
x3 =
6
5
f)
y =
3.Evaluate without using a calculator;
a)
13
 15
d)
5
10
32
g)
3
82 =
13
=
3
2
b)
25 =
c)
2
Key
1. a) x
=
1
3
b)
a)4 n
b)
3. a) -15 b) 125 c) 16
11
17
6
1
3
c) 3 x8
1
(8) 2 =
1
2 3
1
52
1
11
d) 1024
d)
c)94
2
5
3
(64) =
4 2 16 4 =
f)
h)
3
2.
1 3
  =
8
e)
2
3
1
3
e)(13x )  13 x
7
2
d )5 453
e)5 5b3
2
3
f )x
1
f)
5
e) 4
f) 4
g) -4
h) 4
y6
MIAP 20S – Notes
Outcome 10I.A.2: Operations on Radicals (Simplifying)
2, 3,
12 is



5 , 7 , etc. are in
form.
because it contains a
(integer) factor.
12  4  3  4  3  2  3
To simplify a radical (also known as writing as a mixed radical):
 Depending on the
of the radical, look for a perfect
hidden in the factors of the
.

the perfect square (cube, etc.) by placing its root in
the radical sign.

any constants in front of the radicand.
 Leave any
without integer roots
(cube, etc.)
of
.
.
Word bank: combine
inside
not
remove
simplest
factors
numbers
square
the radicand.
front
radicand
square
Examples:
1. Simplify (express as a mixed radical) each radical:
a)
8
d)
3 18 
b)

e)
75 
10 32 
c)

f)
2. Express each mixed radical as an entire radical:
3
a)
b)
3.5 3 
2 5

Homework:
12
Page76, Q #
3
54 
2 3 24 
c)

2 3 4 