Download Grade 9 Mathematics Unit #1 – Number Sense Sub

Grade 9 Mathematics
Unit #1 – Number Sense
Sub-Unit #2 - Powers
Lesson
1
2
3
Topic
Writing Numbers as
Powers
Writing Number Using
Powers of 10
Negative Signs
Associated with the
Base of a Power
“I Can”
 Identify the difference between exponents and bases
 Express a power as repeated multiplication
 Express standard form numbers in expanded form




4
5
6
Zero and Negative
Exponents
Adding and Subtracting
Powers
Exponent Laws









7
Quiz #3 – Powers
Order of Operations



Quiz #4 – Order of
Operations


Explain the role of brackets when the negative sign is
outside the brackets
Explain the role of brackets when the negative sign is
within the brackets
Explain the role of negative signs when raised to a
positive power
Explain the role of negative signs when raised to a
negative power
Identify the value of any expression with an exponent
of zero
Express powers with negative exponents as fractions
Determine the sum of given powers
Determine the difference of given powers
Evaluate an expression using product exponent law
Evaluate an expression using quotient exponent law
Explain why bases must be the same to apply product
or quotient exponent law
Evaluate an expression using power of a product
exponent law
Evaluate an expression using power of a quotient
exponent law
Demonstrate your understanding of above concepts
Solve problems including integers using order of
operations including powers
Solve problems including rational numbers using order
of operations including powers
Identify errors in an incorrect solution
Demonstrate your understanding of the order of
operations
Lesson #1 – Writing Numbers as Powers
Define the word POWER (you may use the glossary in your text book):
Use the power of 62 for the following:
Definitions:
 Exponent – is the raised number using in a power to show the number of repeated
multiplications of the base. In the power above the ______ is the exponent.

Base – The number used as a multiplier for repeated multiplication. In the power above the
______ is the base.

Power – definition above (you wrote it).

Perfect Square – (look up definition in glossary) –
A perfect square can be represented by a box, with the sides of the box
indicating the square root and the number of squares within the box being the
square number. Meaning 2 squared is 4 and the square root of 4 is 2.

Perfect Cube – A number that can be written as a power with an integer
base and exponent of 3. Ex ; eight is a perfect cube number.
Three cubed is eight.
A perfect cube can be represented by a cube made of smaller cubes: This
cube has side length of 2 all around and is composed of 8 cubes.
Ex) Identify the base and exponent in the following:
a) 53
c) x5
b) 107
d) 5t
There are three ways to write powers: As a power, as a repeated multiplication and in standard
form.
Power
32
Repeated Multiplication
3x3
4x4x4x4x4
Standard Form
9
16
125
3
6
7x7x7x7
121
Converting standard form to powers:
**Remember that a power refers to the base position**
To convert a number to a power you need to factor it down using the power specified; this will
mean writing it in expanded form, then determining the exponent.
Ex1) Write the following numbers as a power of 2:
a) 8
c) 32
b)4
d) 16
Ex 2) Write the following standard form numbers as a power of 3:
a) 9
c) 27
b) 81
d) 243
Assignment: Practice Page 55 #4, 7abc, 8abc, 9abc, 11, 12abcd, 13abc
Lesson #2 – Writing Numbers Using Powers of 10
Writing numbers as powers of 10 is also known as writing numbers in expanded form. This
requires knowledge of the place values each digit represents.
For example, the place values in the number 54 627 are as follows:
Digit
Place Value in Words
Place value in
numbers
Place value in
Scientific Notation
5
4
6
2
7
From this we can determine the expanded form of 54 627 as:
Two different methods of expanding numbers are the use of a chart (Method 1) and expanding
(Method 2)
Method 1: Using a values chart
Expand 2865 using a chart:
Thousands (103)
Hundreds (102)
Therefore expanded version is:
Method 2: Expanding
Expand 8593:
Tens (101)
Ones (100)
Ex) Write the following in expanded form using a chart:
a) 4532
b) 453
Ex2) Write the following in expanded form by using the method of expansion.
a) 87962
b) 4587
Assignment: Page 61#6, 8-10
Lesson #3 – Negative Signs Associated with the Base of a Power
How to evaluate powers with negative signs on the base:
There are four possibilities to evaluating powers with negative signs:
1. (-3)4 In this case the base of the power is ________ because the brackets include the
negative. This means that the equation is telling us to do repeated multiplication with -3.
Expanded form:
Ex1) Expand: a) (-4)3
b) (-1)4
**The sign of a ____________ with an ____________ number of negative multipliers is
_____________(if the exponent is even, the product is ___________); the sign of a
________________with an ________________ number of multipliers is _________________ (if
the exponent is odd, the product is ________________).
2. -34 In this case there is an “invisible one” hidden in the number. This really should be
written as
. This makes the base _______. The negative must be included in the
brackets in order to be applied to the base.
Expanded form:
Ex2) Expand: a) -43
b) -14
3. –(-3)4 This case the base includes the negative sign because it is in the brackets.
Expanded form:
Ex 3) Expand: a) –(-4)3
b) –(-1)4
4. –(-34) In this case the negative is not included within the brackets since the exponent is also
within them.
Expanded form:
Ex 4) Expand: a) –(-43)
b) –(-14)
Ex) Complete the table below.
Power
Repeated
Multiplication
2
-2
2
(-3)
(-24)
-(-5)2
(-34)
-(-1)6
-52
(-4)3
Assignment:
Page 55 # 9def, 12ef, 13defghi, 14defgh, 18a
Standard
Form
Lesson #4 – Zero and Negative Exponents
Expand the following:
Exponential
Form
25
24
23
22
21
20
2-1
Expanded Form
2-2
2-3
2-4
2-5
When the exponent is a natural number (remember, like a counting number: 1, 2, 3, …) we
multiply the base by itself the number of times indicated by the exponent.
When the exponent is zero we divide the base by itself, thus always getting a simplified
version of 1.
When the exponent is negative we continue dividing by the base. Another way to look at this
is reciprocating the base to create a positive exponent.
Zero Exponent Law: states that any base raised to the power of _______ is equal to ________.
Ex1) Simplify the following:
a) 52
b) 5-2
c) 50
d) 3-1
e) 4-3
f) 20
g) ( )
h) 3-2
i)
j) 1.5-2
k) 7-2
l) ( )
Assignment: Copy and work on loose leaf in assignments section!
1a. 095
1b. 0.5-1
1c.
2a. 1-41
2b. (-0.6)2
2c. 0.30
3a. (-0.3)3
3b. (-0.3)-2
3c.
4a.
4b. 0.92
4c.
1
-3
-2
0
1d. 0.40
1e.
2d. (-0.7)-2
2e. (-3)0
3d.
3e.
4d.
2
-4
5
-2
4e. (-0.6)0
Lesson #5 - Adding and Subtracting Powers
When evaluating (or simplifying) the sum or difference of a power you must remember to apply
the order of operations (BEDMAS). This means that the power must be applied and simplified
before the sum or difference is evaluated.
Ex 1) Simplify 33-24.
Work
Thought:
Expand power by multiplying the base by itself the number of times
indicated by the exponent.
Simplify exponent by multiplying.
Add or subtract simplified exponents.
Ex 2) Evaluate the following expression:
a)
b)
c)
d) (
e)
(
)
Assignment: page 66 #3abcd, 5abcd, 6a, 7, 8ace, 10bdf.
f)
)
Lesson #6 – Exponent Laws
 Use the pattern below to establish the Product Exponent Law:
Ex 1) Expand
to write as a single power
Numerically:
Explanation:
Original Expression
Expanded form of the powers
Since multiplication is a commutative operation meaning
the order it is performed in doesn’t matter we can remove
the brackets.
We had 2 multiplied by itself seven times; thus the base is 2
and the exponent is 7.
Ex 2) Expand the following to write as a single power
a)
b)
The Product Exponent Law: When multiplying powers with the same base _________ _______
_________________.
 Use the pattern below to establish the Quotient Exponent Law:
Ex 3) Expand
to write as a single power.
Numerically
Explanation:
Original Expression
Expanded expression with the base on each level multiplied
by itself the number of times indicated by the power.
Cancel out “like terms” from numerator and denominator in
the correct ratio.
We were left with 5 multiplied by itself twice, so five
squared.
Ex 4) Expand the following to write as a single power:
a)
b)
The Quotient Exponent Law: When dividing powers with the same base ___________ _____
_____________.
****THESE LAWS ONLY ARE IN EFFECT IF THE BASES ARE THE SAME****
Ex 5) Expand
and use it to establish the Product of a Power Law.
Numerically
Explanation:
Original Expression
Innermost exponent expanded.
is now the base for
the next expansion.
Outermost exponent expanded.
Brackets may be removed because multiplication is
commutative.
The base (5) is multiplied by itself 6 times, so 6 is placed as
the exponent.
Ex 6) Expand the following to write as a single power:
a)
b)
The Power of a Product Exponent Law: When a power is raised to another power ___________
_____ _____________.
Ex 7) Write each of the following as a single power:
a)
b)
c)
d)
e)
f)
(
)
Assignment: Page 76 #4aceg, 5aceg, 8ab; Page 84 #6, 14abcd.
Lesson #7 – Order of Operations
Use BEDMAS to solve problems including powers with negative and positive exponents.
Ex1) Evaluate
[
]
[
Apply BEDMAS
Expand exponents to evaluate
Simplify exponents
Perform multiplication
Perform addition
]
Ex2) Simplify:
a)
b)
[
Assignment: Page 76 # 10ab, 13ab; Page 84 #16abc, 19abc; Page 88 #14ace
]