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Powers and
Exponent Laws Unit 1
Powers When an integer, other than 0, can be written
as a product of equal factors(multiplied by itself), we can write the integer as a power.
For example, 4x4x4 is 43
4 is the base.
3 is the exponent.
43 is the power.
43 is a power of 4.
We say: 4 to the 3rd, or 4 cubed
Writing powers
Write as a power.
3x3x3x3x3x3
7
36 The base is 3. There are 6 equal factors, so the exponent is 6.
71 The base is 7. There is only 1 factor, so the exponent is 1.
Evaluating Powers Write as a repeated multiplication and in
standard form. (Use calculators for standard form)
35
74
64
35 = 3x3x3x3x3, 243
74 = 7x7x7x7, 2401
64 = 6x6x6x6, 1296
Evaluating Expressions
Involving Negative Signs
Identify the base of each power, then
evaluate the power.
(-3)4
-34
-(-3)4
Answers on next slide.
Evaluating Expressions
Involving Negative Signs
Answers:
(-3)4 The base is -3, (-3)x(-3)x(-3)x(-3), The
sign of a product with an even number of
integers is positive = 81
-34 The base is 3, -(3x3x3x3), = -81
-(-3)4 The base is -3, -(-34 ) = -(-81)
Powers of Ten
and the Zero
Exponent September 18, 2017
Before we move on…
(-3)3 = (-3)(-3)(-3) =
What if we think about that another way?
(-3)3 = 1x (-3)(-3)(-3) = -27
This is what is happening whenever we
use exponents!
Based on that knowledge…
What is (3)0 going to be?
Ask yourself, in the power (3)0 how many
times am I multiplying 1 by (3)?
Your answer should be … 0 times.
So what are you left with?
Just the 1!
In other words: (-3)0 = 1
Examples: Solve in your notes!
(-6)2 =
(2)0 =
(8)4 =
(-2)0 =
(10)4 =
You have 4 minutes. You can use calculators.
Answers:
(-6)2 = 36
(2)0 = 1
(8)4 = 4096
(-2)0 = 1
(10)4 = 10 000
Zero Exponent Table
Please complete the Zero Exponent
Activity.
Zero Exponent Law
A power with an integer base, other than
0, and an exponent of 0 is always equal
to 1.
n0 = 1, when n ≠ 0
Remember that ≠ means ‘does not equal’
Powers of Ten Powers of Ten follow a consistent pattern that
help with our intuition and understanding of powers and exponents. Observe.
1x10 = 10
1x10x10 = 100
1x10x10x10 = 1000
Follow this pattern 5 more times.
Powers of Ten This table shows the decreasing powers of
10.
Number in Words
Standard Form Power
Hundred Million 100 000 000 108
Ten Million 10 000 000 107
One Million 1 000 000 106
One Hundred
Thousand
100 000 105
Ten Thousand 10 000 104
One Thousand 1 000 103
One Hundred 100 102
Ten 10 101
One 1 100
What
pattern(s)
do you
see in this
table?
Writing number using powers
of 10
Re-write 3522 using powers of 10.
Use a place-value chart.
Another way: 3522 = 3000 + 500 + 20 + 2
= 1 x (3 x 1000) + (5 x 100) + (2 x 10) + (2 x 1)
= 1 x (3 x 103) + (5 x 102) + (2 x 101) + (2 x 100)
Thousands Hundreds Tens Ones
3 5 2 2
Examples: Solve in your notes!
Write the following numbers using powers
of ten (like the last slide):
246
5702
23005
Examples: Solve in your notes!
Write the following numbers using powers
of ten (like the last slide):
246 = (2x 102)+(4x 101)+(6x100)
5702 = (5x103)+(7x102)+(2x100)
23005 = (2x104)+(3x103)+(5x100)
Questions
Please complete the following questions
from pages 61- #4-11
Order of
Operations with
Powers September 22, 2017 When you see this symbol (**) it means you take down the notes.
Skill-Testing Question: 6 x (3 + 2) – 10 ÷ 2
Which answer is correct: 5, 10, 15, or 25?
How do you know?
6 x (3 + 2) – 10 ÷ 2
= 6 x (5) – 10 ÷ 2
= 30 – 5
= 25
A reminder… **
To avoid getting inconsistent answers when
we solve an expression we use this order of
operations:
BEDMAS or
Evaluate the expressions in brackets first.
Evaluate the powers.
Multiply and divide, in order, from left to right.
Add and subtract, in order, from left to right.
Adding and Subtracting with
Powers **
Solve each expression in your notes.
Let’s do the first one together…
a) 33 + 23
b) 3 – 23
c) (3+2)3
Adding and Subtracting with
Powers **
a) 33 + 23
b) 3 – 23
c) (3+2)3
Adding and
Subtracting
with Powers September 25, 2017
Adding and Subtracting with
Powers
Please practice with the following
examples (make sure to show each step):
1. (23 + 34) – 12 =
2. 73 - (6 + 5 – 4) =
3. (-5)2 + (33 – 25)3 =
4. (7 + 8)2 + (12 – 3)3 =
5. 92 - (7 – 4)3 =
Answers:
1. (23 + 34) – 12
= (8 + 81) -12
= 89-12
= 77
2. 73 - (6 + 5 – 4)
= 73 - (7)
= 343 - 7
= 336
Answers (cont’d):
3. (-5)2 + (33 – 25)3
=(-5)2 + (8)3
= 25 + 512
= 537
4. (7 + 8)2 + (12 – 3)3
= (15)2 + (9)3
= 225 + 729
= 954
Answers (cont’d)
5. 92 - (7 – 4)3
= 92 - (3)3
=81 - 27
= 54
Let’s check your
understanding…
Why are the answers to 33 + 23 and (3 + 2)3 be different? Explain.
Answer: The answers are different because in the 1st expression you cube the numbers first, then add them which equals 35. In the 2nd expression, you add the numbers first, then cube their sum which equals 125.
Multiplying and Dividing with
Powers **
When multiplying and dividing with
powers we must make sure to check the
bracket placement and shape - ( ) or [ ]
For example: [2 x (3)2 – 6] 2 is a lot different
than 2 x (3)2 – 62
Multiplying and Dividing with
Powers
[2 x (3)2 – 6] 2
What do we do first?
Then…
Then…
Finally…
Let’s try these together:
2 x 32
(2 x 3)2
1+5 x 81÷9
(1+5) x 81÷9
2 x 4 + 12
( 2 x 4 + 1)2
For you to attempt (5 minutes): Use each of the digits 2, 3, 4, and 5 once to write an
expression. The expression must have at least one power (xy).
The base of the power can be a positive or negative digit
The expression can use any of: addition, subtraction, multiplication, division, and brackets.
Solve the expression. Then get a partner to solve your expression.
My example: (4 x 5)2 – 3 = (20)2 - 3 = 400 – 3 = 397
Questions…
With all of these please make sure that
you write down the question first and then
show each step
Page 66 #3-5 (a-f), 8, 14, 16
Exponent
Laws 1
A Quick Refresher…
A product is the result of multiplying two or
more numbers.
A quotient is the result of dividing one
number by another.
When we…
Multiply numbers does the order in which
we multiply matter?
NOPE!
For example, (3x3)x3 = 3x(3x3)
We usually just write the product without
brackets: 3 x 3 x 3
Next…
You will be working with your table
partner.
Your task will be to investigate multiplying
and dividing powers.
You will need 3 dice. 2 white dice and
one that is a different colour.
Use this number as
the base
Use these
numbers as
the exponents
Directions:
Roll the dice. Use the numbers to create
powers. (Remember that your blue or red
dice are your base)
Record each product or quotient of powers.
When calculating quotient please put the
greater exponent in the numerator place.
Express each power as repeated
multiplication, and then as a single power. Fill
in your chart completely.
Let’s connect it!
As the dice activity showed us, patterns
are revealed when we multiply and divide
powers with the same base.
To multiply 63 x 64 = (6x6x6)x (6x6x6x6)
= (6x6x6x6x6x6x6)
= 67
To divide 68 ÷ 64 = See example on board
Exponent Law for a Product of
Powers **
To multiply powers with the same base,
add the exponents.
am x an = am+n
The variable a is any integer, except 0.
The variable m and n are any whole
numbers.
Why don’t we just calculate
the power into standard form?
That works for this: 63 x 64
But how could we do it for a3 x a4?
We can’t! The use of variables complicate
things but the law that we just learned will
apply to powers with variables for bases
or exponents.
Exponent Law for a Quotient
of Powers **
To divide powers with the same base,
subtract the exponents.
am ÷ an = am-n m ≥ n
a is any integer, except 0; m and n are any
whole numbers.
Try these: (just solve to the
power)
56 x 52 =
(-3)12 ÷ (-3)6 =
Questions(these will be
checked):
Pages 76-77 # 4,5, 6a, 7-10, 13, 17, 18
Exponent
Laws
Let’s review briefly… What is the law when multiplying powers with
a same base?
You always simplify by adding the exponents.
What is the law when dividing powers with a same base?
You always simplify by subtracting the exponents.
Do you always need to solve the remaining product?
Only if asked to solve.
What do you think happens
here?
(24)3
= 24x3 = 212
As repeated multiplication?
(24)x(24)x(24) = 212
As a product of factors? (Combined
repeated multiplication)
(2x2x2x2)x(2x2x2x2)x(2x2x2x2) = 212
**Exponent Law: Power of a Power: If you are raising a Power to an exponent, you multiply the
exponents!
So, when I
take a
Power to a
power, I
multiply the
exponents
n
m mnx x
62323 55)5(
What do you think happens
here?
(2x4)3
= 23x43
As repeated multiplication?
(2x4)x(2x4)x(2x4) = 23x43
As a product of factors? (Combined
repeated multiplication)
(2x2x2)x(4x4x4) = 23x43
**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.
n n nxy x y
So, when I take a Power of a Product, I apply the exponent to all factors of the product.
222 45)45(
What do you think happens
here?
As repeated multiplication?
As a product of factors? (Combined
repeated multiplication)
3
6
5
3
3
6
5
6
5
6
5
6
5
666
555
**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 n
n
x x
y y
So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.
81
16
3
2
3
24
44
Textbook Q’s:
Pages 84-85: 4,5,6, 8, 10, 14, 16(a,b,c),
17(a,b,c).