Download Unit 2: Powers and Exponent Laws

Unit 2: Powers and
Exponent Laws
Unit 2: Powers & Exponents Intro Activity
How many times can you fold
an 8.5 x 11 piece of paper in
half?
What about an 11x17 piece of
paper?
Unit 2: Powers & Exponents Intro Activity
Create a table like this in your notebook:
Number of Folds
Number of Layers
Number of Layers
as a Power
Use a piece of 8.5 x 11 paper to complete the first two columns.
Once you have done that, look for a pattern and see if you can
complete the third column.
Predict the number of layers if you could fold your paper 25
times
Unit 2: Powers & Exponents Intro Activity
Measure the thickness of 100 pages in
your math or science textbook.
Use number to calculate the thickness of
1 sheet of paper.
How high would the layers be if you could
make 25 folds? Give your answer in as
many different units as you can.
What do you know that is approximately
this height or length?
2.1 What is a Power?
A power is a product of equal factors
23 = 2 x 2 x 2 = 8
54 = 5 x 5 x 5 x 5 = 625
62 = 6 x 6 = 36
Power
2.1 What is a Power?
WARNING! WARNING!
53
DOES NOT MEAN
5x3
2.1 What is a Power?
A power with an exponent of 2 is a
square number
4
2
A power with an exponent of 3 is a
cube number
4
3
2.1 What is a Power?
Expressions using negative signs
When there are no brackets, the
exponent belongs to the integer only:
-43 = - 4 x 4 x 4 = -64
-54 = - 5 x 5 x 5 x 5 = -625
When the exponent is outside a set of
brackets, it belongs to everything inside
the brackets.
(-4)3 = (-4) x (-4) x (-4) = -64
(-5)4 = -5 x -5 x -5 x -5 = 625
2.1 What is a Power?
Evaluate the following:
(-6)4
-64
-(-64)
2.1 What is a Power?
Homework:
4ac, 5ac, 7bdf, 8bdf, 9bdf, 10, 11, 12bdf, 13bdfh,
14bdef, 15b, 16bdf, 18a
2.2 Powers of 10 and the 0 Exponent
Consider 107 ...
What is the base?
What is the exponent?
How do we write it as a repeated
multiplication?
How do we write it in standard form?
Why would a scientist use the power instead of
standard form in a research paper?
2.2 Powers of 10 and the 0 Exponent
Powers of 10 are used extensively
in science and other fields.
109 = 1 000 000 000
108 =
100 000 000
107 =
10 000 000
106 =
1 000 000
105 =
100 000
100 =
What pattern do you notice?
2.2 Powers of 10 and the 0 Exponent
Powers with an exponent of 0 are
always equal to 1:
90 =
-90 =
(-5)0 =
-(-2)0 =
2.2 Powers of 10 and the 0 Exponent
Standard numbers can be written
using powers of 10:
4356 = (4 x 1000) + (3 x 100) + (5 x 10) + (6 X 1)
= (4 x 103 ) + (3 x 102 ) + (5 x 101 ) + (6 x 100 )
63, 708 =
2.2 Powers of 10 and the 0 Exponent
Homework:
4ac, 5ac, 6ae, 8ace, 9cf, 10bf, 11,
13ac,
15 (you may use your phones)
2.3 Order of Operations with Powers
Please Answer the following skill testing question to
claim a prize. Keep your answers confidential, and
submit them to Mr. Denham
6 x (3+6)3 - 10 ÷ 2 + 104 =
Solution: 6 x (3+6)3 - 10 ÷ 2 + 10 4
= 6 x (9)3 - 10 ÷ 2 + 104
= 6 x 729 - 10 ÷ 2 + 10 000
= 4374 - 10 ÷ 2 + 10 000
= 4374 - 5 + 10 000
= 4369 + 10 000
= 14 369
2.3 Order of Operations with Powers
Please Answer the following skill testing question to
claim a prize. Keep your answers confidential, and
submit them to Mr. Denham
6 x (3+6)3 - 10 ÷ 2 + 104 =
Solution: 6 x (3+6)3 - 10 ÷ 2 + 10 4
= 6 x (9)3 - 10 ÷ 2 + 104
= 6 x 729 - 10 ÷ 2 + 10 000
= 4374 - 10 ÷ 2 + 10 000
= 4374 - 5 + 10 000
= 4369 + 10 000
= 14 369
2.3 Order of Operations with Powers
Using the following solution, try to develop a list of "rules" for
which calculations need to be done in what order...
Solution: 6 x (3+2)3 - 10 ÷ 2 + 10 4
1. Calculate stuff in brackets
= 6 x (5)3 - 10 ÷ 2 + 104
= 6 x 125 - 10 ÷ 2 + 10 000
= 750 - 10 ÷ 2 + 10 000
= 750 - 5 + 10 000
= 745 + 10 000
= 10 745
2. Calculate Powers
3. Calculate Multiplications and
Divisions
4. Calculate Additions and
Subtractions
2.3 Order of Operations with Powers
A useful acronym for order of
operations is BEDMAS:
B - Brackets: complete all operations inside of brackets.
E - Exponents: complete all operations involving powers
D - Complete all Divisions and Multiplications from left to right
MA - Complete all Additions and Subtractions from left to right
S
2.3 Order of Operations with Powers
Evaluate:
(3 + 4)2
(16 ÷ 22 + 1)3 + 60
(32 + 5) + 23
72 + [3 + (4+2)2 ] - 5
2.3 Order of Operations with Powers
When you have an expression with a numerator and
denominator, the fraction bar is like brackets... solve the top and
the bottom separately first and then divide.
(4 - 3)2 + 119
(42 + 5) + √81
Homework: 3egh, 4egh, 5egh, 6b, 7, 10acef, 12, 15,
16cd (just solve), 19
2.4 Exponent Laws I
When we multiply numbers, the order in which we multiply does
not matter.
Ex. (2 x 2) x 2 =
2 x (2 x 2) =
....so we usually write the product without brackets ex. 2 x 2 x 2.
Tr
t
I
y
!
- Create a group of _______
- Each group needs three dice
- Create the two tables found on
page 73 in your textbook
Instructions:
1) The first table is for THE PRODUCT of POWERS (Part 1). The
second table you drew is for QUOTIENT of POWERS.
2) Choose which die is going to be for the base, and which two dice
are going to represent the powers. Roll all three dice. Write your two
powers from your roll of the dice and complete your table.
3) Roll the dice 5 times for table 1 and 5 times for table 2
2.4 Exponent Laws I
Example: Product of Powers:
I first roll a ____. My 2nd and 3rd dice are ___ and ___.
Example: Product of Quotients:
I first roll a ___. My 2nd and 3rd dice are ___ and ___.
Your turn . . .
What do you notice?
2.4 Exponent Laws I
We have already learned two exponent "laws:"
Exponent Law #1: a0 = 1
Exponent Law #2: a1 =
Exponent Law #3: a
m
ex. 23 x 210 =
ex. 5 x 53 =
ex. (-4)6 x (-4)4 =
xa =a
n
m+n
where a ≠ 0 and
"m" and "n" are
whole numbers
2.4 Exponent Laws I
What did you notice when you divided powers with the
same base? Can you write exponent law #4?
Exponent Law #4: a
m
ex. 55 ÷ 52 =
ex. 62 ÷ 67 =
ex. (-4)10 ÷ (-4)2 =
÷a =a
n
m-n
where a ≠ 0 and
"m" and "n" are
whole numbers
2.4 Exponent Laws I
Evaluate:
3 2 x 34 ÷ 3 3
(-10)4 [(-10)6 ÷ (-10)4 ] - 107
2.4 Exponent Laws I
52 x 58
58 ÷ 52
x
410 ÷ 43
45
P. 77 # 1, 3, 4acegi, 5gh, 7, 8ace, 10acegi, 12,13cd
2.5 Exponent Laws II
What is the standard from of (23 )2 ?
Power
Product of
Powers
Expanded Form
Simplified
Power
(23)2
Can you write "Law #5?"
Standard Form
(Answer)
2.5 Exponent Laws II
When a power is raised to another power,
you can multiply the exponents.
Exponent Law #5: (am )n = am x n
Write as a single Power:
ex. (23 )2 = (23 ) x (23 ) = 26
ex. (63 )6 =
ex. -(24 )5 =
where a ≠ 0 and
"m" and "n" are
whole numbers
2.5 Exponent Laws II
ex. (3 x 4)5 ..... Any suggestions?
= (3 x 4) x (3 x 4) x (3 x 4) x (3 x 4) x (3 x 4)
=3x4x3x4x3x4x3x4x3x4
=3x3x3x3x3x4x4x4x4x4
= 35 x 4
Can you write law #6?
Exponent Law #6: (ab)m = am bm
ex. (3 x 2)6 =
ex. (32 x 22 )3 =
where a ≠ 0 and
"m" and "n" are
whole numbers
2.5 Exponent Laws II
(
5
6
3
5x5x 5
53
=
=
6
6
6
6
(
Exponent Law #7:
a n
=
(
(
b
Write as a quotient of Powers:
84
((
13
=
an
bn
3
where a ≠ 0 and
"m" and "n" are
whole numbers
Examples:
Evaluate:
a) (32 x 33 )3 - (43 x 42 )2 =
c) [(-5)3 + (-5)4 ]0 =
b) (6 x 7)2 + (38 ÷ 36 )3 =
Practice P. 84,85 # 4cd, 5cd, 6, 8ace, 11, 14aceg, 15, 17ace, 19de