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).