Download Unit 9 NCP 504 506 Worksheet

NCP 504, 506 Worksheet CRS SKILL Numbers: Concepts & Properties NCP 504 NCP 506 Name_________________________________________ Period____________ LEVEL Level 1 – ALL students must attain mastery at this level DESCRIPTION NCP 401 Exhibit knowledge of elementary number concepts including rounding and ordering decimals Level 2 – MOST students will NCP 504 Work with scientific notation attain mastery of the focus skill in NCP 506 Work with problems involving positive isolation . integer exponents Level 3 – SOME students will attain mastery of focus skill with other skills Level 4 – SOME students will attain mastery of focus topics covered in a more abstract way Level 5 – FEW students will NCP 604 Apply rules of exponents attain mastery of the extension skill. Level 1 1. Calculate each of the following. (Without a calculator) b. 8 − −5 = c. −8 + −5 = a. 8 + −5 = d. −8 − −5 = e. 8 ÷ −5 = f. 8 ∗ −5 = g. −8 ∗ −5 = h. −8 ÷ −5 = −8
i. = −5
2. Complete each by filling in the missing space. !
!
23. 4× ⋯×4 = b. ! × ⋯× ! = ! !"#$%
!" !"#$%
4 × 4 × 4 × 4 × 4 × 4 × 4 = c. 3.6× ⋯×3.6 = 3.6!" _______ !"#$%
e. 𝑥 ∙ 𝑥 ⋯ 𝑥 = !"# !"#$%
!
!" !"#$%
f. 𝑥 ∙ 𝑥 ⋯ 𝑥 = 𝑥 ! 1 !
d. − !" × ⋯× − !" = _______!"#$%
3. Will these products be positive or negative? How do you know? a. −1 × −1 × ⋯× −1 = −1
!"
b. −1 × −1 × ⋯× −1 = −1
!" !"#$%
!"
!" !"#$%
4. Is it necessary to do all of the calculations to determine the sign of the product? Why or why not? a. −5 × −5 × ⋯× −5 = −5
!" !"#$%
!"
b. −1.8 × −1.8 × ⋯× −1.8 = −1.8
!"" !"#$%
5. Fill in the blanks about whether the number is positive or negative. 1. If 𝑛 is a positive even number, then −55 ! is __________________________. b. If 𝑛 is a positive odd number, then −72.4 ! is __________________________. 6. Josie says that −𝟏𝟓 × ⋯× −𝟏𝟓 = −𝟏𝟓𝟔 . Is she correct? How do you know? 𝟔 𝒕𝒊𝒎𝒆𝒔
7. Rewrite each number in exponential notation using 2 as the base. 2 8 = 16 = 64 = 32 = 128 = 256 = !""
8. Tim wrote 16 as −2 ! . Is he correct? 9. Can you find a number n such that (-­‐2)n = 32? How about (−2)! = 64? Why or why not? 10. Solve the problems below. Look for a pattern. 8×10 = 8 ÷ 10 = 8×10×10 = 8 ÷ 10 ÷ 10 = 8×10×10×10 = 8 ÷ 10 ÷ 10 ÷ 10 = 8×10×10×10×10 = 8 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = 8×10×10×10×10×10 = 8 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = a. What happens when you multiply by 10 each additional time? b. What happens when you divide by 10 each additional time? Level 2 11. Are the following numbers written in scientific notation? If not, state why. a. 1.908×10!" b. 4.0701 + 10! c. 0.325×10!! d. 18.432×5! 3 e. 7.99×10!" f. 8×10!!! 12. Write each in scientific notation a. 7,300,000 b. 8,000,000,000 c. 53, 450 d. 34,000,000,000,000 13. Write each in standard notation a. 5.62×10! 14. Write each in scientific notation a. 0.000973 b. 3.602×10! c. !−9.3×105 c. !5653×10−9 c. !9.02×10−6 b. -­‐0.0034 15. Write each in standard notation a. !3×10−4 b. !1.692×10−7 16. There are approximately 7,300,000,000 people on earth as of September 1, 2014. Write this number in scientific notation. 17. Compare 9.3×10!" and 9.2879×10!" using > or <. Explain your reasoning. 18. Chris said that 5.3×10!" < 5.301×10!" because 5.3 has fewer digits than 5.301. Show that even though his answer is correct, his reasoning is flawed. Show him an example to illustrate that his reasoning would result in an incorrect answer. Explain. 4 19. If stacked perfectly 3.906×10! ping-­‐pong balls should fit into an Olympic swimming pool. a. Write the number of ping-­‐pong balls in standard notation. b. If the mass of one ping-­‐pong ball is 2.7 grams, what is the mass of all ping-­‐pong balls that would fill an Olympic swimming pool? c. If an average male gorilla weighs 330 pounds, how many gorillas would it take to match the weight of the ping-­‐pong balls filling the swimming pool? 20. Simplify the following using rules of exponents a. !x 3 x 10 b. !x 2 y 7 ⋅ x 5 y 6 58
2
!5
x 12
f. 6 !x
i. x 5 !
j. 2x 2 !
e. ( )
2
5 ( )
5
( )( )
c. 5x 7 8x !
x 10 y 12
g. 5 5 !x y
d. (!6x 2 )(!4x 2 ) h.
k. !(4p2 )3 20x 5 y 2
2
4x
y
!
l. !(−2t 4 f 7 )4 21. Simplify the following using rules of exponents a. !(x 3 )2 b. !(x 3 )(x 2 ) 2 3
7
5
e. !(8n u )(−2n )(4u ) ( )( )
3
f. −9 −9 !
c. (5 p 3 )(−m 8 p 2 ) 2
3
g. (8n p) 4x 18 y 8
i. 3
20x
y
!
j. v7
2
!9v
k. !2x 4 +5x 4 d. !42 ⋅44 910
h. 6 !9
l. !4(5t 7 )2 22. Simplify the following using rules of exponents 3
2
5
a. (3!x y )(-­‐6 y ) (
3
8
)
4
2
b. 2 x
!
e. (10 g 3 h 8 v 6 )(11gh 8 ) f. −22 x 3 y 4 (−3)2 x 4 y 4 !
h. !3x 3 y 5(4x 2 y 6 )2 6 (
⎛ 4a 3 ⎞
c. ⎜ 4 ⎟ ⎝ 2b ⎠
)(
i. !(4d 4n7 )5 )
12xy 7x 5 y 2
⋅
d. 4
4
y
7x
!
g. !(xy)2(x 2 y 2 )2 j. !(−2x 5 y 8 )4 23. Solve for the missing variables a. If !(15)5 = 5t ⋅3t , what is t? b. If !(85 )4 = 8r , what is r? Level 3 24. Multiply or divide each number in scientific notation without using a calculator. Write your final answer in scientific notation. 8 ×105
a. !(3×104 )(2×106 ) b. !(6 ×105 )(4 ×109 ) c. 2
!2×10
25. The mass of water on the Earth’s surface is 1.41 x 1018 metric tons. The total mass of the Earth is 5.97 x 1021 metric tons. Express the mass of water on the Earth’s surface as a percentage of the total mass of the Earth. 26. All planets revolve around the sun in elliptical orbits. Uranus’s furthest distance from the sun is approximately 3.004×10! km, and its closest distance is approximately 2.749×10! km. Using this information, what is the average distance of Uranus from the sun? 27. Here are the masses of the so-­‐called inner planets of the Solar System. Mercury: 3.3022×10!" kg Earth: 5.9722×10!" kg Venus: 4.8685×10!" kg Mercury: 6.4185×10!" kg What is the average mass of all four inner planets? Write your answer in scientific notation. 7 Use the table below to complete the next four problems. Population State Debt (in dollars) (2012) California 407,000,000,000 38,000,000 New York 337,000,000,000 19,000,000 Texas 276,000,000,000 26,000,000 North Dakota 4,000,000,000 69,000 Vermont 4,000,000,000 62,600 Wyoming 2,000,000,000 57,600 28. The table below shows the debt of the three most populous states and the three least populous states. a. What is the sum of the debts for the three most populous states? Express your answer in scientific notation. b. What is the sum of the debt for the three least populated states? Express your answer in scientific notation. c. How much larger is the combined debt of the three most populated states than that of the three least populous states? Express your answer in scientific notation. 29. What is the sum of the population of the 3 most populous states? Express your answer in scientific notation. 30. What is the sum of the population of the 3 least populous states? Express your answer in scientific notation. 31. Approximately how many times greater is the total population of California, New York, and Texas compared to the total population of North Dakota, Vermont, and Wyoming? 8 32. Americans make almost 2 billion telephone calls each day. a. Write this number in standard notation and in scientific notation. b. How many phone calls do Americans make in one year? (Assume that there are 365 days in a year.) Write your answer in scientific notation. 33. The speed of light is 300,000,000 meters per second. The sun is approximately 1.5×10!! meters from Earth. How many seconds does it take for sunlight to reach Earth? 34. The mass of the moon is about 7.3×10!! kg. It would take approximately 26,000,000 moons to equal the mass of the sun. Determine the mass of the sun. 35. The mass of Earth is 5.9×10!" kg. The mass of Pluto is 13,000,000,000,000,000,000,000 kg. Compared to Pluto, how much greater is Earth’s mass? 36. Using the information in the last two problems, find the combined mass of the moon, Earth, and Pluto. 37. How many combined moon, Earth, and Pluto masses are needed to equal the mass of the sun? 9 38. The sun produces 3.8×10!" joules of energy per second. How much energy is produced in a year? 39. On average, Mercury is about 57,000,000 km from the sun, whereas Neptune is about 4.5×10! km from the sun. What is the difference between Mercury’s distance and Neptune’s distance from the sun? 40. Find the area of each figure a. b. c. 41. The area of a rectangle is !30x 6 y 8 . If the with of the rectangle is !5xy 6 , what is the height? Level 4 42. Simplify each expression a. 10 ! !
!
! !
!
= b. (3𝑥 ! 𝑦 ! 𝑧 ! )(−𝑦 ! 𝑧 ! ) c. !(−4x 6 y 8 z 2 )3(8x 3 y 4 z 10 )2 (
2
)
4
⎛
⎞
43. Simplify ⎜ 3x 6 y 10 ⎟ ⎠
!⎝
44. Find the missing variable value a. If !(83 )2 = 2h , what is the value of h? b. If !(272 )6 = 3n+2 , what is the value of n? 45. Show that the mass of a proton is 0.938 GeV/c2 using the information to the right. Gigaelectronvolt, GeV/c2, is what particle physicists use as the unit of mass. 1 Gigaelectronvolt = 𝟏. 𝟕𝟖𝟑 × 𝟏𝟎!𝟐𝟕 kg Mass of 1 proton = 𝟏. 𝟔𝟕𝟐 𝟔𝟐𝟐 ×𝟏𝟎!𝟐𝟕 kg 46. The distance of the nearest star (Proxima Centauri) to the sun is approximately 4.013 336 473×
10!" 𝑘𝑚. Show that Proxima Centauri is 4.2421 light-­‐years from the sun. 11 In popular science writing, a commonly used unit is the light year, or the distance light travels in one year (note: one year is defined as 365.25 days). 𝟏 𝒍𝒊𝒈𝒉𝒕 𝒚𝒆𝒂𝒓 = 𝟗, 𝟒𝟔𝟎, 𝟕𝟑𝟎, 𝟒𝟕𝟐, 𝟓𝟖𝟎. 𝟖𝟎𝟎 𝒌𝒎 ≈ 𝟗. 𝟒𝟔𝟎𝟕𝟑×𝟏𝟎𝟏𝟐 𝒌𝒎 47. The Fornax Dwarf galaxy is 4.6×10! light-­‐years away from Earth, while Andromeda I is 2.430×
10! light-­‐years away from Earth. Which is closer to Earth? Explain your reasoning. 48. The average lifetime of the tau lepton is 2.906×10!!" seconds and the average lifetime of the neutral pion is 8.4×10!!" seconds. Explain which subatomic particle has a longer average lifetime. 49. A rectangular area of land is being sold off in smaller pieces. The total area of the land is 2!" square miles. The pieces being sold are 8! square miles in size. How many smaller pieces of land can be sold at the stated size? Compute the actual number of pieces. 50. Let A and B be two positive numbers, with no restrictions on their size. a. Is it true that 𝐴×10!! < 𝐵×10! is always true? Explain. b. Now if 𝐴×10!! and 𝐵×10! are written in scientific notation, is it always true that 𝐴×10!! < 𝐵×10! ? Explain. 12 !
51. A certain ball is dropped from a height of 𝑥 feet, it always bounces up to ! 𝑥 feet. Suppose the ball is dropped from 10 feet, complete the table below indicating the distance the ball bounces after each bounce. Height of the ball after the indicated Bounce number of bounces (Use exponents) Height of the ball after the indicated number of bounces in standard notation 1 2 3 4 30 𝑛 52. The same ball is dropped from 10 feet and is caught exactly at the highest point after the 15th bounce. What is the total distance traveled by the ball? Use what you learned from the last problem. 53. The mass of a neutron is approximately 1.674927×10!!" kg. The mass of a proton is 1.672622×
10!!" kg. Explain which is heavier. 54. The average lifetime of the Z boson is approximately 3×10!!" seconds and the average lifetime of a neutral rho meson is approximately 4.5×10!!" seconds. a. Explain why the neutral rho meson has a longer average lifetime. b. Approximately how much longer is the lifetime of a neutral rho meson than a Z boson. 13 Level 5 ⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞
⋅
⋅
⋅
⋅
55. Circle all of the following that are equivalent to ⎜⎝ −8 ⎟⎠ ⎜⎝ −8 ⎟⎠ ⎜⎝ −8 ⎟⎠ ⎜⎝ −8 ⎟⎠ ⎜⎝ −8 ⎟⎠ !
5
−5
!(−8) 5
⎛ 1⎞
− ⎜ ⎟ ⎝ 8⎠
!
⎛ 1⎞
⎜⎝ − 8 ⎟⎠ !
−5
!8 56. Categorize the value of each exponential expression Less Between 0 Greater than -­‐1 and 1 than 1 −8
!5 3
!(−5) −4
!(−5) −2
!(5) 6
!(−5) 57. Simplify each expression. Write your final answer with positive exponents only 22
x5
x −1 y
2 5 −2
−4 0
a. −9 b. !(4x y ) c. !−3x y d. −2 e. −2 !2
!x
! xy
5x 3 y 9
(5m)−2
f. g. h. !(x 3 )0 2 −2
−8
!20x y
! 7p
14 i. !(x 4 y 7 )−3 j. !(10x 5 y 3 )−3 ( y 8 )−3
k. 8 9 !( y )
58. Simplify each expression. Write your final answer with positive exponents only −1
4x 3 y 3 5xy 2
⋅
a. 2xy
2
y
!
36a8b2 ⎛ 6 ⎞
b. ⋅
ab ⎜⎝ ab2 ⎟⎠
!
4
16x 5 y −8 ⎛ x 3 y 2 ⎞
c. 7 4 ⋅ ⎜
⎟ x y
⎝ 8xy ⎠
!
(
3
5x −3 y 2 2xy
e. 5 −1 ⋅
xy
!x y
)
(
)
−2
2
6x −2 y 2 4x y
d. ⋅
−3
xy
xy 2
!
⎛ 2xy −2 y 4 ⎞ ⎛ 4xy ⎞
f. ⎜
⎟ ⋅
−1
−1 −3
⎝ 3x y ⎠ ⎜⎝ 2x y ⎟⎠
!
8m3p9
10m8 p
b. ÷ 3 −6 3 2 2
!(4m p ) m p
−2
−2
59. Complete each division problem 7x 2 y 6 14x 6 y 8
a. 2 7 ÷
3 0
!9x y 21x y
Review 60. Simplify each expression. 15 a. −6n + 5(4n + 1)
b. −2x + 3(x − 5)
c. −6(1 − 8x) − 6
d. 5x + 3(2x − 8) 2
61. Complete each conversion. c. 4.875 miles = ______
16 yards
d.
6600 yards = _____ miles
e.
7920 inches = _____ meters
f.
3755 ounces = ______ gallons