Download Irrational numbers

BELL RINGER 3/27
•SOLVE FOR X:
2X + 7 = -23
X = -15
3X – 4 = 11
X=5
OBJECTIVE
• REVIEW EXPONENT RULES AND RATIONAL NUMBERS
• GUIDED PRACTICE
• DAILY CLASS ASSESSMENT
MGSE 8.EE.3
Exponents
5
exponent
3
Power
base
Example: 125  53 means that 53 is the exponential
form of the number 125.
53 means 3 factors of 5 or 5 x 5 x 5
The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
x  x  x  x  x  x  x  x
n
n times
n factors of x
Example: 5  5  5  5
3
#2: Multiplying Powers:
If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
x x  x
m
So, I get it!
When you
multiply
Powers, you
add the
exponents!
n
mn
2 6  23  2 6  3  29
 512
#3: Dividing Powers: When dividing Powers with the same base, KEEP
the BASE & SUBTRACT the EXPONENTS!
m
x
m
n
mn

x

x

x
n
x
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
6
2
62
4

2

2
2
2
 16
#4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
x 
n
m
So, when I
take a Power
to a power, I
multiply the
exponents
x
mn
32
(5 )  5
3
2
5
5
#5: Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result is a
multiplication of individual factors of the product, each powered
by the given exponent.
 xy 
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
n
x y
n
n
(ab)  a b
2
2
2
#6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
n
 x
x
   n
y
 y
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
n
4
16
2 2
   4 
81
3 3
4
#7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
x
m
1
 m
x
1
1
5  3 
5
125
and
1
2

3
9
2
3
3
#8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
x 1
0
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.
50  1
and
a0  1
and
(5 a ) 0  1
SOLUTIONS
2 2
1. 3  3 
2
2.
2
3
a a  a
5
2
5 2
3
4
a
3. 2s  4s  2  4  s
2
7
4. (3)  (3)  (3)
2
5.
3
s t s t 
2 4
7 3
s
7
27
23
 8s
 (3)
2 7 43
t
9
s t
5
9 7
SOLUTIONS
12
7.
8.
9.
10.
s

4
s
9
3

5
3
12 8
s t

4 4
st
5 8
36a b
4 5
4a b
s12 4  s 8
395  34
s
12 4 8 4
t
s t
8 4
5  4 85
3
36

4

a
b

9
ab

SOLUTIONS
1. 3
2. a

2 5



3 4
3. 2a

2 3
4. 2 a b
2
10
3
a 12
3
 2 a

5 3 2
23
 8a
6
 2 22 a 52b 32  2 4 a10b 6  16a10b 6
5. (3a )   3  a
22
6. s t
6 12
2
2 2

2 4 3
23 43
s t
s t
 9a
4
SOLUTIONS
5
s
7.   
t
5
s
5
t
2
3 
8.  5   34
3 
9
 
2
3
8
2
4 2
2 8
 st 


st
s
t
9.  4   
  2

r
 rt 
 r 
8
 36a b
10 
4 5
4
a
b

5 8
2

  9ab3



2
2 32
9 a b
2
 81a b
2 6
SOLUTIONS
1. 2a b   1
0
2
 
1
3. a
 5
a
5
2
7
4. s  4s  4s
5 1
5. 3x y
2
6. s t

3 4

2 4 0

4
 3 x y
 1
8
12

8
x

81y12
SOLUTIONS
1
2 
7.  
 x
9 2
3 
8.  5 
3 
2
1
x
4
 x   4
 
 
 3
2
4 2

1
3  8
3
8

s t 
 2  2 2
4 4
s t
9.  4 4   s t
s t 
10
2
5
b

2

2
10
 36a 
9
a
b

2


10.  4 5  
81a
4
a
b


2 2
Rational and Irrational
Numbers Essential Question
How do I distinguish between
rational and irrational numbers?
Make a Venn Diagram that displays the following sets
of numbers:
Reals, Rationals, Irrationals, Integers, Wholes, and
Naturals.
Reals
Rationals
2
3
-3
-2.65
Integers
-19
Wholes
1
0
6
Naturals
1, 2, 3...
4
Irrationals

2
Recall that rational numbers can be written as
the quotient of two integers (a fraction) or as
either terminating or repeating decimals.
3
4
= 3.8
5
2
= 0.6
3
1.44 = 1.2
Irrational numbers can be written only as
decimals that do not terminate or repeat. They
cannot be written as the quotient of two
integers. If a whole number is not a perfect
square, then its square root is an irrational
number. For example, 2 is not a perfect square,
so 2 is irrational.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.
A fraction with a denominator of 0 is undefined
because you cannot divide by zero. So it is not a
number at all.
Check It Out! Example 1
Rational, Irriational, Not a Real Number
1.
9
9
=3
rational
2.
3.
–35.9
–35.9 is a terminating decimal.
rational
81
3
rational
81
9
=
=3
3
3
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
4.
21
irrational
5.
0
3
rational
0
=0
3
Additional Example 2: Determining the
Classification of All Numbers
State if each number is rational, irrational,
or not a real number.
6. 4
0
not a real number
Check It Out! Example 2
State if each number is rational, irrational,
or not a real number.
7.
23
23 is a whole number that
is not a perfect square.
irrational
8.
9
0
not a real number
Check It Out! Example 2
State if each number is rational, irrational,
or not a real number.
9.
64
81
rational
8
9
8
64
=
9
81