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?


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


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

Power

23 = 2 x 2 x 2 = 8

54 = 5 x 5 x 5 x 5 = 625

62 = 6 x 6 = 36


53 DOES NOT MEAN

5 x 3

WARNING! WARNING!


A power with an exponent of 2 is a square number

A power with an exponent of 3 is a cube number

42

43

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)


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?


Powers of 10 are used extensively in science and other fields.

109 = 1 000 000 000108 = 100 000 000107 = 10 000 000106 = 1 000 000105 = 100 000

What pattern do you notice?

100 =


Powers with an exponent of 0 are always equal to 1:

90 = -90 = (-5)0 = -(-2)0 =


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 =


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


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


Solution: 6 x (3+2)3 - 10 ÷ 2 + 10 4

= 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

Using the following solution, try to develop a list of "rules" for which calculations need to be done in what order...

1. Calculate stuff in brackets

2. Calculate Powers

3. Calculate Multiplications and Divisions

4. Calculate Additions and Subtractions


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

M -

A - Complete all Additions and Subtractions from left to right

S


Evaluate:

(3 + 4)2 (32 + 5) + 23

(16 ÷ 22 + 1)3 + 60 72 + [3 + (4+2)2 ] - 5


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

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.

- 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

Try It!

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 ___.

What do you notice?

Your turn . . .

2.4 Exponent Laws I

where a ≠ 0 and"m" and "n" are whole numbers

ex. 23 x 210 =

ex. (-4)6 x (-4)4 =

ex. 5 x 53 =

2.4 Exponent Laws I

We have already learned two exponent "laws:"

Exponent Law #1: a0 = 1

Exponent Law #2: a1 =

Exponent Law #3: am x an = am + n

where a ≠ 0 and"m" and "n" are whole numbers

2.4 Exponent Laws I

Exponent Law #4: am ÷ an = am - n

ex. 55 ÷ 52 =

ex. (-4)10 ÷ (-4)2 =

ex. 62 ÷ 67 =

What did you notice when you divided powers with the same base? Can you write exponent law #4?

Evaluate:

32 x 34 ÷ 33 (-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

2.4 Exponent Laws I

What is the standard from of (23 )2 ?

Can you write "Law #5?"

2.5 Exponent Laws II

Power Product of Powers Expanded Form Simplified

PowerStandard Form (Answer)

(23)2

ex. (23 )2 = (23 ) x (23 ) = 26

When a power is raised to another power, you can multiply the exponents.

ex. (63 )6 =

Write as a single Power:

ex. -(24 )5 =


Exponent Law #5: (am )n = am x nwhere a ≠ 0 and"m" and "n" are whole numbers

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)= 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4= 3 x 3 x 3 x 3 x 3 x 4 x 4 x 4 x 4 x 4= 35 x 4

Can you write law #6?

ex. (3 x 2)6 = ex. (32 x 22 )3 =


Exponent Law #6: (ab)m = am bmwhere a ≠ 0 and"m" and "n" are whole numbers

ab( (n an

bn=

56

3 56 x=

53

63

813( (4

=

Write as a quotient of Powers:

( (2.5 Exponent Laws II

56 x 56 =

Exponent Law #7: where a ≠ 0 and"m" and "n" are whole numbers

Examples:Evaluate:

a) (32 x 33 )3 - (43 x 42 )2 =

b) (6 x 7)2 + (38 ÷ 36 )3 =

c) [(-5)3 + (-5)4 ]0 =

Practice P. 84,85 # 4cd, 5cd, 6, 8ace, 11, 14aceg, 15, 17ace, 19de

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Unit 2: Powers and Exponent Laws