1. Exponents and Scientific Notation.
In this unit we will explore how we apply the properties, and laws of exponents, to solve problems, of which some will involve numbers written in scientific notation.

2. POD
Evaluate the expression x4 if x = 4, then try it when x = -4

3. POD
Evaluate –x2 if x = 3
Then try again, if x = - 3

4. Peasant Chessboard Question
A peasant in an ancient kingdom, is walking through the royal forest. All of a sudden, he spies a robbery taking place. With out hesitation, the brave peasant stops the robber! After tying the suspect to a tree, he realizes the woman being robbed was the princess. To thank the peasant, the king invited the peasant to the castle, and offered him a great reward. The king told the peasant, “I can never truly repay your bravery in saving my only daughter, but I will give you 1 million dollars as a token of my gratitude”. The peasant however said, “thank you sire, but I am a humble man, and instead would gladly take a mere 1 dollar today on one square of the royal chessboard. The next day, I will take two dollars on the next square, and the day after that, four dollars on the following square. Each day, all you have to do is double what was on the previous square, until all the squares on the chessboard have been filled.”
Which reward plan would you rather receive? The kings plan or the peasant’s plan?

6. Because exponents are how we think about, and write, the repeated multiplication of the same number, we can use an exponential expression to show the repetition of “doubling” in the peasant’s plan.

7. Determine what word, or idea, can sensibly fill in the blank, so to complete this analogy, and then justify your response: “Multiplication is to addition, as __________________ is to multiplication.”
(Understanding analogies: An analogy is a comparison of similarities, between things that otherwise are dissimilar. For example, we could say a boat is to water, as a plane is to what?

8. Multiplication: The REPEATED ADDITION of the same number.
When studying exponents in algebra, it is important to reacquaint ourselves with some definitions which will inform how we simplify expressions, and understand properties of exponents.
Multiplication: The REPEATED ADDITION of the same number.
Thus, instead of we write 6(4)
Or, instead of x + x + x + x + x + x we write 6x
Exponents: The REPEATED MULTIPLICATION of the same number.
Thus, instead of 4(4)(4)(4)(4)(4) we write 46
Or, instead of x(x)(x)(x)(x)(x) we write x6

9. Why do we use contractions, and slang, like…
Aight
Don’t
No prob
Can’t
I’d
Should’ve
…Instead of…?
Alright
Do Not
No Problem
Can Not
I would
Should have
This is a bit like why we use exponents

10. How does this graph represent a comparison between linear and exponential functions or expressions?

11. x3 or 5x3 Vocabulary Review
How do we refer to the different parts of an exponential expression? Why is it important that we all use the same vocabulary?
x3 or 5x3
Exponent, or power. A “2” we can called “squared”, and 3 we can say is “cubed”. Or we just say “x to the power of 3.” The power tell you how many bases are multiplied together
When a number is left of a variable, even if that variable is raised to a power, we call it a “coefficient”. This means that 5 is MULTIPLIED by whatever the value is, of x3
The Base:
The base is the number which is multiplied by itself.

12. In this unit we will be using algebraic expressions which involve exponents. We will expand and simplify the way different expressions are written in order to better understand how to write and interpret these expressions.
Expanding: expanding an expression is when we write an expression, in its LEAST simplified form.
Eg #1: 6d = ?
The coefficient of “6” means that 6 is being multiplied by “d”, or that there are “six d’s” added together.
= d + d + d + d +d + d
Eg #2: d6 = ?
Because the “6” is a power, this does not mean 6*d. Rather, the power tells us that there are “six d’s” multiplied together.
= d*d*d*d*d*d

13. Eg #3: How do we expand an expression with a coefficient and a power?
We can either expand the coefficient…
= d4 + d4 + d4
…Or we can expand the power…
= 3 * d * d * d * d
…Or we can fully expand both the coefficient and the power…
= d*d*d*d + d*d*d*d + d*d*d*d

14. Simplifying: When we can simplify an expression, it usually means that we can write an equivalent expression, which uses the fewest amount of symbols, and operations.
Eg #1: d + d + d + d = 4d
Eg #2: 3f + f + f  Sometimes to simplify, we will fully expand the expression first…
f + f + f + f + f = 5f
Eg #3: f*f*2*f*3*f  Sometimes we reorder the numbers and variables, using the commutative property of multiplication or addition.
2*3*f*f*f*f = 6f4
Eg #4: f*f*f + g + g + f*f = f3 + 2g + f2
We can not combine the f3, and the f2, because of the
“+” sign between them.

15. Write in most simplified form
Write in expanded form
“Cubby” Groups: do all of set A and C
“Middle” Groups: do at least 2 problems from each set, A, B, C and D.
“Window” Groups: do all of set B and D. B)
1.) 5k - k 2.)(h3)(h2)
3.) x +2x +x
4.) 5x + 2y
A) 1.) 3k 2.)k3 3.) x +2x +x 4.) 5h – 2h
Write in most simplified form D)
5.) L + L + L
6.)2h + h + h
7.) x*2*x*x*5*x
8.) f*f*f + 5g*g
9.) y + 3y + y – y + y2 C)
5.) k + k ) (k)(k)
7.) x + x + x +x
8.) 2*f*f
9.) 3y*y + y

16. POD Try to write each in their most simplified form.
Only one can be simplified any further.
If you’re stuck expand the entire expression before simplifying.
Why can one of these be simplified further, while the other can not?
1.) J3 + J ) H3 + H3

17. POD: Simplify. Rewrite with one base and one power
POD: Simplify. Rewrite with one base and one power. If done early, try number 2.
1.)C5 * C-2 =
2.) C-3*C5 * C-2 =

18. How can we understand the first (1st of a few we will learn) laws of exponential expressions?
But first some background…
Take these notes down in your notebook, on some basic principles of exponents:
X0 = 1  That means that any number base, to the zero power, will always represent a value of 1.
Examples: 560 = 1, = 1, P0 = 1, (xyz) 0 = 1
X1 = X  That means that any number base to the power of 1, is always itself.
 The reverse is also true  X = X1  Which means that any number or variable written without a visible exponent, is said to have “an invisible 1” in the power. Many times throughout this unit, you may find it helpful to write the power of one, when it is not already written.
Examples: 51 = 5, 831 = 83, M1 = M,
(93abc)1 = (93acb)

19. The 1st Product Law of Exponents
(AB)(AC) = A(B+C)  If like bases are multiplied the expression can be rewritten, as that base, to the sum of the powers.
Examples:
1.) 57*53 = 510, because 57*53 = 5(7+3)
2.) 46 * 43 * 45 = 414,
because 46 * 43 * 45 = 4(5+3+6)
3.) x6 * x = x7 , because x6 * x1 = x(6+1) (notice the “invisible” 1 as the power of x.

20. Why does it work to add the exponents when multiplying?
Sometimes to help us to expand an expression, before simplifying, to better understand how it is works.
Ex1.) 57*53 =
5*5*5*5*5*5*5*5*5*5
So because 7 fives being multiplied, is being multiplied by another 3 fives being multiplied, its all just multiplication of the same number a total of 10 times. Thus, we can simplify by adding the exponents for the total number of times 5 is being multiplied. So our answer is 510

21. Lets practice simplifying a few expressions together
34*33 =
d2*d4 =
2r5*3r2 =

22. Practice simplifying exponential expressions
Practice simplifying exponential expressions. Rewrite the expression with as few bases as possible. If you are stuck, try expanding the expression first, and don’t forget when you can apply the commutative property of multiplication or addition. A
1.) Y2Y5
2.) 63(62)
3.) Y2Y
4.) 3Y2Y9
5.) G2H2G7
6.) 5(52 )(56)
7.) Y2X2X7Y
8.)3d2ad
9.) 63(6-2)
10.) B
1.) Y2Y(Y3 )
2.) 63(62)
3.) 3Y2(7.5)Y9
4.) B2B2B
5.) Double what 23 is.
6.) 5(5-2 )(56)
7.) Triple what 37 is.
8.) 3Y2X2(5Y4X)
9.) Y2Y + Y2Y
10.) 3Y3Y + 2Y3Y

23. 1st Quotient Law of exponents.}
AB ÷ AC = A(B-C) What this law means, is that when two like bases are divided, the quotient can be written as that base, to the difference of the powers.
AB = A(B-C) AC
Examples:
76 = 7(6-4) = 72 74
6X9Y12 = 3X2Y4
2X7Y8

24. Practice one together…
Rewrite with only one base and power
581 53

25. Practice one on your own/with your group, to check your understanding…
Rewrite with only one base and power.
x16 x7

27. Practice applying the 1st quotient law: rewrite the following expressions with as few bases and powers A.
1.) 58 53
2.) (-x)9
(-x)8
3.) U19
U -4
4.) 18X6Y8
4X3Y3
5.) 8x8 3x3
12x3
1.) 58 53
2.) 49 42
3.) U19 U4
4.) X6Y8
X3Y3
5.) 35x8x2
5x3

29. Power Law of Exponents. (5x4)3 = (51*3)(x4*3) = (53x12)
(Ab)c = Ab*c  A power raised to a power, is equal to that same base raised to the product of those powers. (AbGf)c = (Ab*c)(Gf*c) When there is more than one base within the parenthesis being raised by the “outside power”, the outside power is multiplied by ALL the “inside” powers. Examples: (32)3 = 36 Because… (32)3 = 32*3
(5x4)3 = (51*3)(x4*3) = (53x12)

33. Lets apply the power law together…
How can we rewrite this example, with one base, and one power?
(J5)-9

34. Do one on your own, in your groups, to check your understanding…
(3-3)-4

35. Power Law Continued… Why? See this example expanded…
(32)3 = (32) (32) (32) = = 36
Because there are 3, two’s being added in the powers, we just multiply them instead.

36. Practice the power law…
A. 1.) (T3)3 2.) (T3)-5 3.) (Y-9)-4 4.) (2T3)4 5.) (F7Y)3 6.) (T3X2)3(T3) B.
1.) (Q6.5)-7
2.) (T-8)-5
3.) (3Y-2)-4
4.) (2T3)4(2T3)
5.) (X)(X5Y3Z2)4
6.) (2X6Y)8
(4X3Y)3

37. Justify how you know what the value of the power “b” must be, in order for the equation below to be true.
(8b)4 = 824

38. Understanding negative exponents
How do we interpret a negative exponent?

39. Understanding negative and zero exponents.
How would we describe what happens, every time the exponent INCREASES by 1?
Therefore, how can we describe what happens as the exponent DECREASES by 1?
How can we explain the pattern when the exponent decreases by 1, by using multiplication?
5*5*5*5*5 55
3125
5*5*5*5 54
625
5*5*5 53
125
5*5 52 25 5 51
5/(5) 50 1
5/(5*5)
5-1
1/5
5/(5*5*5)
5-2
1/(52) or 1/25
5/(5*5*5*5)
5-3
1/(53) or 1/125 ÷5 *5 ÷5 *5 ÷5 *5 ÷5 *5 ÷5 *5 ÷5 *5 ÷5 *5 ÷5 *5

40. Rewriting negative exponents by finding the reciprocal.
Review: Reciprocal: When a number is written as a fraction its reciprocal is when we switch the numerator with the denominator. This is known as the multiplicative inverse, because when a number multiplies by its reciprocal, the result will always be 1.
Ex1
Ex2
6 * = 1 6
2 * = = 1
6 * = 1

41. Prior knowledge connection
How did you learn to divide by fractions?
Ex ÷ 3_ = 4
Keep, change, “flip”  means find reciprocal
6 * 4_ = = 8

42. Find the reciprocal 73 X5

43. Examples of how we rewrite expressions with negative exponents, by finding the reciprocal.
 This means that any base raised to a power, which is negative, means that the expression can be rewritten as the reciprocal of that base to that power.
5-1 = 1_ 51
 Because we do not leave expressions written with negative exponents, we must rewrite the value as the positive exponent’s reciprocal, using a fraction.
6y5x-67 = 6y5_
x67
 Only reciprocate the base, which has the negative exponent.

45. Meet Mr. (or Mrs.?) Negative Exponent.
People really don’t get me. People think I’m such a downer, but I swear I’m really not that negative. I don’t make the number negative at all actually. I just remind the mathematicians to put the base and power under “1” in the denomenator.

48. Simplify expressions making sure to rewrite expressions containing negative exponents, so there are no negative exponents. All rewritten expressions should be in furthest simplified exponential form. Any written responses are in complete sentences.
Ex: X9/X11 = X9-11 = X-2 = 1/(X2)
Group A:
1.) 3-3 Hint: Find the reciprocal of 33 2.) X ) 25/ ) X-5(X3) 5.) 5X-9X2
6.) Create an expression with no negative exponents, that when simplified, has a negative exponent. How did you know this expression would result in a negative exponent?
Group B:
1.) Why are these expressions not equivalent? ≠ -27
2.) X3/X9 3.) 6X3/(X5)4 4.) [X3(X5)]/ (X3) ) 21X-9YX2
Group C:
1.) A classmate insists that negative exponents mean that the value of the expression is negative, or to the left of zero on the number line. Justify that the value of an expression with a negative exponent lies to the right of zero on the number line. Effectively make your case!
2.) 3X3(2X5)/ (6X3) ) 20X-4/5X ) 7X-5/21X ) 20(X-5)3/3X8

49. Exponents Properties Project:
You will be assigned a station.
Each station has 5 columns. You choose one problem from each column.
From the first four columns, rewrite the expression given with as few bases and powers as possible, leaving no negative exponents in your final representation.
The final column, you must answer in complete sentences. However showing and referring to mathematical work, may greatly assist your understanding.
You may represent you project in a flip book, a foldable, on loose-leaf, or on white paper.

50. Rubric
4: I understand the problems. My answer is correct. I applied an appropriate rule, concept, or process, and verified my strategy is correct. In my written response, I used a lot of specific math language, coherently communicating a clear conceptual understanding. All of my math thinking is correct.
3: I understand most of the problems fully. My answers are correct, or incorrect due to careless errors rather than conceptual misunderstanding. I at least attempted to apply the appropriate rule. In my written response, I used some math language accurately. All of my math thinking is correct, but could have gone further to elaborate concepts.
2: I only understand parts of the problems, and am sometimes applying an appropriate strategy. In my written response, I used some math language, but could be more clear and specific. Some of my math thinking is correct.
1: I do not understand the problems. In my written response I used no math language, and did not clearly convey a conceptual understanding. My math thinking is not correct.

51. Name:___________________________________________Class:___________ Exponent Short Assessment: Simplify the expressions in #’s 1 – 4, leaving them in exponential form, using the properties and laws of exponents which have learned about. Leave no final answers with negative exponents. Any expressions with negative exponents must be rewritten.
1.) 45(43) ) 7G6(3G4)
3.) 16X10
2X2
4.) X3
X15

52. Scientific notation: Why is it used
Scientific notation: Why is it used? How do we understand how it is used?

53. POD
Which factor is more important when considering how large the quantity is, that is being represented? The left-factor, or the right-hand factor? Use these two numbers written in scientific notation as an example:
9.08(102) and 1.1(1034)

54. Understanding scientific notation.
How do we understand ways to exactly rewrite a number from standard notation, into scientific notation, or back the other way?
See three board examples, one standard, one scientific, and one neither, but looks like sci. not.
How do we compare numbers written in scientific notation?
How can we apply…
multiplication to numbers written in S.N.?
Division to numbers written in S.N.?
Addition to numbers written in S.N.?
Subtraction to numbers written in S.N.?

55. Observing the difference between different representations of the same number. Lets look at two different numbers…
Words
8.5 Trillion
35 Trillionths
Standard Notation
8,500,000,000,000
Scientific Notation
8.5*10^12
(^ means “to the power of”)
3.5*10-11
Neither Standard or Scientific Notation. (please avoid this)
85*10^11
0.35*10-10

56. Examples of scientific notation in use.
Diameter of our solar system: approximately 9 billion km, or 9*10^9
Mass of a single water molecule: approximately 3*10^-23

57. What’s the problem we want to avoid in the last row?
Scientific notation is a quantity represented by two factors…
The left factor: must be greater than or equal to 1, and less than ten. (between 1 and ten, or equal to one) (1 ≤ (left factor) < 10)
The right factor: must be a power of ten. (a base of ten raised to some power)

58. Why do we need to know and understand scientific notation
You may go into science or engineering…
Large or small answers displayed on calculator.
More efficient ways to express very large or very small quantities.
Why do we need a more convenient way of expressing extremely small quantities?

59. Lets model a couple more
1.) Std  Sci.
A.)
B.) 56,000,000,000
2.) Sci  Std.
A.) 1.5x10^5
B.) 1.5x10^-5

60. You try… 1.) Std  Sci. 2.) Sci  Std. A.) 0.0000074 B.) 90,000,000
A.) 1.6x10^6
B.) 8.5x10^-12

61. Comparing numbers written in scientific notation.
Listing them from least to greatest.
Strategy Option 1:
Double check that they are all in correct scientific notation.
Order all the numbers from the smallest POWERS of ten, to the greatest POWERS of ten.
For any numbers with the same power of ten, then order those numbers by the size of the LEFTHAND FACTOR.
Strategy Option 2:
Convert all numbers into standard notation.
Order them from least to greatest.
If directions necessitate, after numbers are in the proper order, convert them all back into scientific notation.

62. Practice with ordering numbers in scientific notation.
Group A: 5x10^5, 3.2x10^5, 1.2x10^7, 3.23x10^5, 5.0x10^4
Group B: 20x10^4, 1.5x10^5, 12x10^6, 8x10^4
Group C: 83,000, 83x10^4, 8.3x10^3, 8030, 8.31x10^4

63. How do we apply multiplication and division to scientific notation?
EX1: 5*103 (3*105)
5*3*(103)*(105)  Regroup so left-hand factors are next to each other, and the powers of ten are next to each other.
15*(103+5)  Multiply left-hand factors together, and multiply powers of ten together. Why can we add the exponents?
15*(108)  Understand why this is not in correct scientific notation.
1.5 *(109)  The left hand factor must be between 1 and 10, so moving the decimal 1 space left, is like dividing by ten, so we must increase the right hand power of ten so it balances out.
EX2: (For student to work out on board)
(7*10^3)(2*10^9)

64. Practice multiplying numbers written in scientific notation to get a product. Be sure and represent all products in correct scientific notation form.
ALL GROUPS: When you finish your group’s problem set, write an explanation of your strategy OR describe exactly what you find to be most difficult about multiplying numbers in scientific notation.
Group A:
1.) (2*10^3)(4*10^6) ) (2*10^5) * 3
3.) 10 * (2*10^5) ) (4*10^7)(3*10^6)
Group B:
1.) 6*10^3(3*10^5)*2 2.) (1.2*10^5)(8*10^3)
3.) .03 * (4*10^3) 4.) 6.7*10^7(2*10^2)
Group C:
1.) 6.3*10^5(3*10^17) 2.) 4*10^5(3*10^-12)
3.) (9*10^6)÷(1.5*10^4) 4.) (7.5*10^5)÷(2.5*10^-5)

65. POD
How can we multiply these numbers, with a product written in scientific notation, without turning them into standard notation first?
4.9x10^5 (3x10^6)

66. How can we apply the same strategies used for multiplying, when we divide numbers written in scientific notation?
Ex1) 3.6(107)
1.2(105)
*Divide the coefficients
*keep the base of ten, SUBTRACT the exponents.
= 3 * (107-5) = 3(102)
You may still use strategy option #2, but by now, you may be finding that it is taking more time than necessary to change every number into standard notation, and then back again.
If you are still confused or intimidated by strategy #1, please use this time to revisit any questions you may have about this strategy.

68. The previous problem, using strategy #2:
3.6(107) = 36,000, = 36,000,000
1.2(105)
= = 3(102) 12
*Please let me know if you would like a review of how to use long division, or other strategies, in order to correctly divide 3,600 by 12.

70. Group A.) Group B.) Group C.)
Practice dividing numbers in scientific notation. All quotients (how would you define quotient?) must be in scientific notation.
Group A.)
1.) 9(106) ÷ 3(103) 2.) 2(107) ÷ 8(103)
Group B.)
1.) 9.6(106) ÷ 1.2(103) ) 9(106) ÷ 3(1012)
Group C.)
1.) [9(106)] ÷ [3(103) * 2(107)]
2.) [ 9(106) ÷ 3(103) ] + 2(103)

71. POD: Divide the numbers in scientific notation below, and keep your quotient in scientific notation
3(106)
1.2(1028)

72. A B C 3.) 3(1015) 1.5(10-10) 1.) 3*(104) 1*(102) 1.) 3*(108) 1.5(102)
More practice with dividing and multiplying quantities expressed in scientific notation. If you finish early, besides other acceptable activities, you may practice more in “Pre-Algebra” p221, and 223. A B C
1.) 3*(104)
1*(102)
1.) 3*(108) 1.5(102)
1.) 1*(108)
3*(1015)
2.) 3(106)*5(1012)
2.) 8(1012)*1.6(1012)
2.) 3.8(106)*1.5(10-12)
3.) 3(1015)
1.5(1010)
3.) 3(1015)
1.5(10-10)
3.) 2(10-15)
1.5(10-19)
4.) 4.5(107)
9(104)
4.) 5.4(10-7)
9(104)
4.) 4.5(107)*-3(106)
9(104)*4(107)
5.) 3(106)
1.5(1010)
5.) 3(106)*8(105)
1.5(1039)
5.) 3(106)3*[3(106)]2
9.9(1039)

73. Scientific notation: “Real World” applications
A passenger airline can hold 3.2*10^2 passengers. If there are roughly 6*10^3 passenger planes in the air worldwide at any given time, approximately how many people are in the skies at any given time?

74. Word Problem Strategies.
Even before I feel I know how to find the solution, what can I do to demonstrate that I am working towards an understanding of the problem?
1.) Close read!
2.) Make the problem simpler: substitute “easier” numbers, then figure out the strategy, and finally, use that strategy with the original numbers given in the problem.
3.) All answers must be given in a complete sentence including any appropriate units.

75. POD
The mass of our sun is 1.98x10^30kg. About how many earths would we need to equal the same mass as the sun, if the earth’s mass is 5.97x10^24?

76. Applying multiplication and division of numbers in scientific notation: All answers must be in scientific notation.
Group A: A company owns 3 warehouses. Each warehouse can hold 5*104 boxes. How many boxes all together can all three warehouses hold?
Group B: A plane travels at a speed of 3*10^3 miles per hour. If the plane travels in a straight line, at that same constant speed, how long will it take this plane to travel 1.2*10^5 miles? (You can use the formula rate*time=distance.)
Group C: A spaceship is travelling to a satellite to repair it. The spaceship is 5*10^12 miles away from the sun. If the satellite is 2*10^5 miles away from the sun, how many times further from the sun is the spaceship than the satellite?

77. Word problem practice (involving scientific notation).
1.) The distance from Pluto to the sun is about 5.9×10 12 meters, which is the radius of our solar system. The Milky Way galaxy has a radius of approximately 3.9×1020 meters. How many times greater is the radius of the Milky Way galaxy than the radius of our solar system?
2.) The body of a person of average weight contains about 2.3 x 10-4 lb of copper. How much copper is contained in the bodies of 1200 such people?
3.) A liter is equal to 1 x 106 mm3. There are roughly 5 x 106 red blood cells in 1 cubic millimeter of human blood. How many red blood cells are there in a liter of human blood (hint: 1 liter = 1,000 ml)?
4.) *challenge* Using the following data: the volume of a sphere = (4/3)(∏)R (For ∏, use 3.14) Radius of the earth= 6.3×106 m. Mass of earth= × kilograms
 Find the density of the earth in kg/m 3

78. “How many times greater is _______ than ________?”
This is a frequent question you may see on word problems involving scientific notation, because scientific notation is so often used to understand the magnitude of things. This question asks us to compare the magnitude of two related quantities as the larger value being a multiple of a certain factor of the smaller value.
What THIS QUESTION IS REALLY ASKING YOU: “What do we need to multiply the smaller value by, to get the bigger value?”
What THIS QUESTION IS NOT ASKING YOU: This question is often confused for “how much greater is _______ than _______?” “How much greater” questions usually ask you to subtract, where “how many times…” questions involve multiplication and division.
What YOU NEED TO DO: Because we don’t know the missing factor the smaller value is multiplied by, we can use the inverse of multiplication, and divide the larger value by the smaller one.
Short example: How many times larger is 48, than 12?
Divide the larger value by the smaller one to find the missing factor we are looking for. 48/12 = 4. So we write, “48 is 4 times larger than 12.”

79. Example of “how many times greater?” questions
The Earth is approximately 9.3 x 107 miles from the sun. If Mercury is approximately 36 million miles from the sun, how many times farther from the sun, is Earth than Mercury?
Convert 36 million into scientific notation, so both quantities are expressed in the same form.
Divide the larger value by the smaller value
Express you answer in a complete sentence including appropriate units.

80. More practice with “Real world” scientific notation problem solving.
A: Denver, Colorado is approximately 1x103 miles from Los Angeles, and New York is approximately 3 x 103 miles from Los Angeles. How many farther from Los Angeles is New York, than Denver?
B: The half-life of “Uranium 234” is 250,000 years and the half-life of Plutonium is 8.0 x 107 years. How many times greater is the half-life of Plutonium than Uranium 234?
C: There are nearly 21,000 runners in the New York City Marathon. Each runner runs a distance of 26.2 miles. If you combined the total number of miles for all the marathoners, how many times the circumference of the earth would this be? Consider the circumference of the earth to
be 2.5 x 104 miles.

81. More Word problem practice…
1.) Light travels at approximately 3.0 x 108 m/sec. How far does light travel in 6 week?
2.) There are nearly 20,000 runners in the New York City Marathon. Each runner runs a distance of 26 miles. If you add together the total number of miles for all runners, how many times around the globe would the marathon runners have gone? Consider the circumference of the earth to be 2.5 x 104 miles.
3.) One of the fastest supercomputers in the world is NEC's Earth Simulator, which operates at a top-end of 40 teraflops (forty trillion operations per second). How long would it take this computer to perform 250 million calculations?
4.) The half-life of “Uranium 234” is 2.5 x 105 years and the half-life of Plutonium is 8.0 x 107 years. How many times greater is the half-life of Plutonium than Uranium 234?
5.) The Earth is approximately 9.3 x 107 miles from the sun. How long does it take light from the sun to reach the earth? Use the speed of light to be 1.86 x 105 miles per second.

82. Scientific notation word problem Rubric
Some notes/annotations are shown, selecting only the most important information
Mathematical steps are thoroughly shown, and represent a successful strategy for solving the problem.
Answer is correct, written in scientific notation
Answer is written as it relates to the problem using appropriate units.
Written explanation of strategy is an accurate summary of a successful problem solving strategy.
Mostly, mathematical steps are shown, and represent a successful strategy for solving the problem.
Answer is correct, written in scientific notation, or incorrect due to 1 careless error
Answer may or may not be written as it relates to the problem using appropriate units.
Mostly, the written explanation of strategy is an accurate summary of a successful problem solving strategy.
Some notes/annotations may or may not be shown, selecting only the most important information
Somewhat, the mathematical steps are shown, and represent a near successful strategy for solving the problem.
Answer is not correct, due to more than 1 careless error, and no more than 2 conceptual errors. It may or may not be written in scientific notation,
notes/annotations may or may not be shown
Mathematical steps may or may not be shown, but do not represent a successful or near-successful strategy for solving the problem.
Answer is not correct
Written explanation of strategy is not an accurate summary of a successful or near-successful problem solving strategy.
4.)
3.)
2.)
1.)

83. POD: Combine like terms.
How many X’s do we have if we add
3x +5x + 11x ?
How would it be different if we tried to add…
3x +5x + 11x2

84. Adding and Subtracting numbers written in Scientific Notation.
Why are we able to add the left hand factors (coefficients) together in example #1, but not so in EX#2?
EX1: 3(10^4) + 4(10^4) = (3+4)(10^4) = 7(10^4)
EX2: 3(10^4) + 4(10^5) ≠ (3+4)(10^4) ≠ 7(10^4)
and ≠ (3+4)(10^9) ≠ 7(10^9)

85. How do we add/subtract numbers in scientific notation?
1.) 7(10^4) + 4(10^4) =
(7+4)(10^4)  Combine coefficients, keep the base and power of ten the same.
11(10^4)  Assess if sum/difference of coefficients, is between 1 and 10
1.1(10^5)  Adjust the size of the coefficient, as we’ve done, such that it is between 1 and 10, but balance the number out by changing the exponent. How do we know to raise or lower the exponent?

86. How do we add/subtract numbers in scientific notation?
2.) 7(10^4) - 4(10^4) =
(7- 4)(10^4)  In this case we are subtracting, so coefficients are subtracted. Keep the base and power of ten the same.
3(10^4)  Assess if sum/difference of coefficients, is between 1 and 10
In this case, the “3”, is between 1 and 10 so we are finished.

87. You try one:
8.3(10^14) + 2(10^14)

88. Set A.) Set B.) 1.) 5(106) + 3(106) 2.) 8(107) - 2(107)
1.) 5(106) + 3(106) 2.) 8(107) - 2(107)
3.) 1*(109) + 3(109) 4.) 4(1013) - 6(1013)
5.) 9.6(105) - 1.2(105) ) 9(106) + 3.1(106)
7.) How many water molecules are in two cups of water if one cup contains about 8.36 x 1024 molecules of water and another cup contains 4 x 1025?
Set B.)
1.) 9.8(1011) + 3(1011) 2.) 3(1015) - 5(1015)
3.) 9.06(108) – 7.8(108) 4.) [9(106) - 3(106)] * 2(107)
5.) [ 9(106)] ÷ [3(103) + 2(103)] 6.) [9(106) - 6(106)]3
7.) 3 cups of water contain approximately 2.4(1025) water molecules. One cup contains about 8.36 x 1024 molecules of water. How many water molecules are there in 1 less cup than 3 cups of water? (How can we alter the exponent of one of the numbers in scientific notation, so that both powers of ten are the same, allowing us to subtract?)

89. How do we add/subtract numbers in scientific notation, when the powers of ten are close, but not equal?
Ex1) 1.23(10^6) + 5.7(10^7)
Step1) change one of the exponents so they are equal. (We can make the bigger one smaller, or the smaller one bigger, but for now, to keep things simple, lets all make them equal by MAKING THE SMALLER EXPONENT BIGGER.)
 ___(10^7) + 5.7(10^7)
Step2) adjust the decimal of the coefficient the same number of times as the exponent had to change. The power only changed by “1” so decimal moves 1 place.
Step2A) move the decimal in the direction so balance is maintained. Because our exponent got larger, the decimal moves to make the coefficient smaller.
 0.123(10^7) + 5.7(10^7)
Step3) add the coefficients, keep the base of ten and its power the same.
 5.823(10^7)
Step4) Assess if the coefficient is too large (greater than or equal to ten), or too small (less than one). This example is finished.
If so, adjust the power of ten and decimal point of coefficient
Make sure balance is maintained, and your final answer is equal to sum or difference from step 3.

90. POD
#1) 3(108) + 4.1(108)
#2) 3(108) – 4.1(107)

91. Group A.) Group B.) Group C.)
Practice adding/subtracting numbers in scientific notation. All answers must be in scientific notation.
Group A.)
1.) 9(106) + 3(105) 2.) 9(106) - 8(106)
3.) 1.2(106) + 3(106) 4.) 4.5(107) - 6(106)
Group B.)
1.) 9.6(106) - 1.2(105) ) 9(105) + 3(106)
3.) 9.06(109) – 9.8(108) ) 9(1016) - 3(1015)
Group C.)
1.) [3(106) - 9(105)] * 2(107)
2.) [ 9(106)] + [3(102) + 9(102)]2

92. Group A.) Group B.) Group C.)
Practice adding/subtracting numbers in scientific notation. All answers must be in scientific notation.
Group A.)
1.) 7(104) + 4(105) 2.) 2(1011) - 8(1010)
3.) 1.3(106) + 3(107) 4.) 7(107) - 6(106)
Group B.)
1.) 9.01(106) - 3.2(105) ) 9.3(106) + 3.9(106)
3.) 7.06(109) – 9.2(107) ) 9(1016) – 3.7(1015)
Group C.)
1.) [8.2(109) (108)] * 4(107)
2.) [ 9.1(1024)] + [1.2(107) (108)]3

93. Practice Converting numbers between standard and scientific notation.
Group A:
1.) 2(10^-7) ) )
4.)
Group B:
1.) ) 34(10^-4) into scientific
3.) 2.402(10^-9) ) [4.5(10^4)]*[3.2(10^-10)]
Group C:
1.) ) 345.7(10^-12)
3.) [3.1(10^-9)]÷[2(10^4)]
4.) Why will it be difficult to write the sum of 8(10^-8) and 3(10^13) in scientific notation? Hint: Try finding the sum, as a start.

94. Group A.) Group B.) Group C.)
Practice dividing and multiplying numbers in scientific notation, that have negative exponents. All quotients and products must be in scientific notation.
Group A.)
1.) 9(10-6) * 3(10-3) 2.) 2(10-7) ÷ 8(103)
Group B.)
1.) 8.6(10-6) * 1.2(103) ) 9(106) ÷ 3(10-12)
Group C.)
1.) [9(10-6)] ÷ [3(103) * 2(10-7)]
2.) [ 9(106) ÷ 3(10-3) ] + 2(103)

95. Scientific notation and place value connection.
Write each number in scientific notation. (Even if it doesn’t seem necessary because the # isn’t that large). When you are done write a description of the pattern you observe.
1.) 30,000
2.) 3,000
3.) 300
4.) 30
5.) 3
6.) 0.3
7.) 0.03
8.) 0.003
9.)

96. For example…
The approximate number of water molecules in a cup of water is , a very large number, which is much easier to represent as 8.36*10^24, because the standard version uses MANY DIGITS.
While that is a very large number, consider how small just one individual water molecule is. The approximate mass of just one water molecule is grams. There must be an easier way *10^-23

97. 0.000034 is 34 millionths, or 3.4 hundred thousandths.
Example of a very small number written in different forms, to better understand the meaning of the negative exponent in scientific notation.
is 34 millionths, or 3.4 hundred thousandths.
3.4(10-5) This is how the above number is written in correct scientific notation. How do we know that the exponent is -5?
3.4 (1/10)(1/10)(1/10)(1/10)(1/10)
3.4 ( )
10*10*10*10*10
3.4 ( )
105

98. Practice with ordering numbers in scientific notation that have negative exponents.
Group A: 5x10-5, 3.2x10^-5, 1.2x10-7, x10-5, 5.0x10-4
Group B: 20x10-4, 1.5x10-5, 12x10-6, 8x10-4
Group C: , 83x10-4, 8.3x10-3, , 8.31x10-4

99. Scientific Notation: Short Assessment.
1.) Change the number in to scientific notation.
2.) Order the following numbers, from least to greatest:
3.4(10^7), 3.04(10^7), 9.999(10^4), 2(10^-6), 3(10^-5)
3.) What is 3(10^7) times 3.1(10^-5)?
4.) What is 2(10^5) divided by 4(10^13)?
5.) What is the sum of 8.3(10^7) and 7.8(10^7)?

100. POD About how many times 4.01*10^6 is 2*10^8?
The numerical answer needs to be in scientific notation.
Hint: Remember to contextualize our answers when answering a ‘word-problem’. Our answer should be written out like…
“2*10^8 is approximately ___________ times 4.01*10^6.”

101. POD
Which expression is twice the value of 215? 230 or 216? EXPLAIN.

102. POD
How do we simplify the following expressions differently? Answer this, then try to simplify each expression. A.) 7x(3x2 – 5x5y) B.) 7x(3x2 * 5x5y)

103. 1.) 2-4/ ) (5j-6) ) 2-7/ 2-4
4.) 3kh(6g7h3) ) 4pj(5p4 + j2)
6.) The speed of a space-shuttle is approximately 5.3*102 meters per second. About how far does the rocket go in 8.7*104 seconds?
7.) If the area of a rectangle is represented by 12x5, and its width is 20x2, what expression represents this rectangle’s length?

104. Here is an alternate list of the properties of exponents.
Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1)
• Exponents are subtracted when like bases are being divided (Example 2)
• A number raised to the zero (0) power is equal to one. (Example 3)
• Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a positive if left in the denominator. (Example 4)
• Exponents are added when like bases are being multiplied (Example 5)
• Exponents are multiplied when an exponents is raised to an exponent (Example 6)
• Several properties may be used to simplify an expression (Example 7)

105. Examples of Laws & Properties

106. Examples of Laws and Properties

107. Practice
Write the following in expanded form. Then Simplify the expression as much as possible.
1) x5
2) 36 (33)
3.) X2 (X4)
4.)
5.) X2 + X2 + X3 + X2
6.) Y(X3)(Y)(X)
7.) 35 33
8.) (23)4
Group A) Write each in expanded form, and then simplify numbers 2, 3 and 7 as best you can.
Group B.) Write each in expanded form, and simplify all of them as best you can.
- Then show (write about) a connection between any of the properties in our notes and the problems to which you think they apply.
Group C) After writing each in expanded form, simplify all of them as best you can. Justify how we can simplify these, or not simplify them, citing a property in our notes.
- Then write…
1.) How does number 5 demonstrate why we can sometimes add or combine terms, and sometimes we can not. How can the order of operations help you justify why this is?

108. How can we apply our expanded form to understand how some of our properties of exponents work?
Group A) Write each in expanded form, and then simplify numbers 2, 3 and 7 as best you can.
Group B.) Write each in expanded form, and simplify all of them as best you can.
- Then show (write about) a connection between any of the properties in our notes and the problems to which you think they apply.
Group C) After writing each in expanded form, simplify all of them as best you can. Justify how we can simplify these, or not simplify them, citing a property in our notes.
- Then write…
1.) How does numbers 4 & 5 demonstrate why we can sometimes add or combine terms, and sometimes we can not? How can the order of operations help you justify why this is?
1.) X*X*X*X*X 2.) 3*3*3*3*3*3 (3*3*3) 3.) X*X (X*X*X*X) 4.) 2*2*2 + 2*2*2 + 2*2*2 5.) X*X + X*X + X*X*X + X*X 6.) Y*Y (X*X*X) 7.) 3*3*3*3*3 3*3*3

109. How can we apply our expanded form to understand how some of our properties of exponents work?
1.) X*X*X*X*X count the number of x’s, that’s your exponent 2.) 3*3*3*3*3*3 (3*3*3) because we have all the same base being multiplied, 6 times, and then 3 times, and those products also being multiplied, you should see how we can just count all the LIKE BASES (the 3’s in this case) for a total of LIKE BASES, giving us an exponent of 9. 3.) X*X (X*X*X*X) you see the same principle at work as in #2 above, only the base is a variable instead of a know number. Why are these properties of exponents particularly helpful when we have a variable as our base? 4.) 2*2*2 + 2*2*2 + 2*2*2 we can’t just add the exponents in this case because the terms are being added. You only add the exponents if you multiply the terms, and the bases are the same. However we can simplify this addition, because we are repeatedly adding the same exact number. We have 3 * 23, or 3 * 8. 5.) X*X + X*X + X*X*X + X*X same principle as #4, but one of these terms can not be added to the others. We do have repeated addition of X2, but not X3 6.) Y*Y (X*X*X) cannot simplify, bases not the same. 7.) 3*3*3*3*3 here we see why we can subtract the exponents if LIKE 3*3*3 BASES are being DIVIDED. Each like base on top can be “canceled out”, with each like base on the bottom, which the lines are signifying. After we a certain number of bases left over. When the “leftovers” are on top, the exponent is positive, which gives us 32

110. Journal Entry
Biologists need math too! A group of biologists were discussing the results of research they had conducted about the local rabbit population. In the beginning of the research the rabbit population was 2^10, and by the end of the year, the population had doubled. The problem is that half of the scientists thought that meant there were now 2^20 number of rabbits, and the other half thought that meant there were now 2^11 number of rabbits. Which Result correctly represents how the population of 2^10 doubled? Justify your conclusion in words, and show any mathematical work which you think helps support your conclusion.

111. Homework.
#1: p , do all quickchecks #1-3 (#1a, 1b, 1c, 2a, etc…)
#2: p186, #1-7
#3: (division)p197, #6-9
#4: p197, #10-14
#5:(power rule) p.199 read examples, do #1-8
#6: p197, #15-21 (odd #’s only)

112. Short assessment on Monday (1/2 period)
Product law of exponents
Quotient law of exponents
Power law of exponents
Rewriting expressions with negative exponents

113. Homework #7: p178-179, read and do quickchecks #1-4
#8: p180, #4-24 even numbered problems only. (4, 6, 8, etc…)
#9: p188, read and do quickchecks
#1(a, b, & c), and #2
#10: p190, #1, 3, 7, 8, 9

114. Homework #11: p.202-203 read, and do Quickchecks 1 - 3
#12: p #15, 20 and p #18, 28, 29
And every night be studying for the unit exam. See testing calendar!