Name ______________

Date _______________

1. QUESTION: Explain how to evaluate 43 without a calculator. The small raised number is called an exponent, and 43 is a power of 4. Write 4 ⋅4 ⋅ 4 ⋅ 4 ⋅4 as a power of 4. Write the product 43 ⋅45 as a power of 4.

ANSWER: 48

b) 183 c) 826 d) 1926

e) 613

Simplify as much as possible.

f) 20 p11q4

g) 4k 4

h) 500= 1

2. QUESTION: (a) The table contains seven measurements written in decimal and scientific notation. . Complete the table so that each measurement is written in both decimal and scientific notation. . In the last column, rank the measurements in order of size.

(1 = smallest, 2 = next smallest, and so on up to 7 = largest)

 

(b) i. Complete the following statement using two numbers in decimal notation from the table.

______________ × 4000 = _______________

. Complete the following statement using two numbers in scientific notation from the table.

______________ × 50,000 = _______________

ANSWER: (a)

 

(b) i. There are two alternatives:

0.01 m x 4000 = 40 m

200 m x 4000 = 800,000 m

(b) ii. There are two alternatives:

8 x 10−¿ 4 ¿ m x 50,000 = 4 x 101 m

4 x 10−¿ 3¿ m x 50,000 = 2 x 102 m

3. QUESTION: Represent each of the following rational numbers in fraction form.(a) 0.333

(b) 0.317

(c) 2.16

ANSWER: The solution for all the parts of this take advantage of the repeating structure of the decimal expansions. Namely, by multiplying by a suitable power of 10 (namely, 10r where r is the length of the repeating segment in the decimal expansion) and subtracting the original number, we can get a multiple of x with a finite decimal expansion.

(a) Let x=0.333 Then

10x=3.33 = 3+0.333 = 3+xx from both sides gives 9x=3, so

0.333 = x = 39 =

13 .

(b) Let x=0.31717 . . .

100x=31.717171 . . .=0.317171. . .subtracting the two equations gives 99x=31.4, so

0.317 = x = 31.499 =

314990 .

(c) Let x=2.166. . . Then

10x = 21.6666 . . .= 2.1666 . . .subtracting the two equations gives 9x=19.5, so

2.16 = x = 19.59 =

19590 .

4. QUESTION: Find the value of (3x104) + (2 x 102) + (4 x 10).

ANSWER: 30240

5. Side lengths are 4 cm each.

6. QUESTION: Exponents are routinely encountered in scientific work, where they help invesigators deal with large numbers:

(a) The human population of Earth is roughly 7000000000, which is usually expressed in scientific notation as 7×109. The average number of hairs on a human head is 5×105. Use scientific notation to estimate the total number of human head hairs on Earth.

(b) Light moves very fast – approximately 3 ×108 meters every second. At that rate, how many meters does light travel in one year, which is about 3 ×107 seconds long? This so-called lig h t year is used in astronomy as a yardstick for measuring even greater distances.

ANSWER:

a) 3.5 ×1015hairsb) 9 ×1015meters

7. QUESTION: Decide whether each of the following numbers is rational or irrational. If it is rational, explain how you know.

(a) 0.333

(b) √4

(c) √2 = 1.414213 . . .

(d) 1.414213

(e) π = 3.141592. . .

(f) 11

(g) 17 = 0.142857

(h) 12.3456565656

ANSWER:

(a) Since

0.333 = 13

0.333 is a rational number.

(b) Since

√4 = 2 = 21

√4 is a rational number.

(c) √2 = 1.414213 . . . is not rational. In eighth grade most students know that the square root of a prime number is irrational as a "fact," but few 8th grade students will be able to prove it. There are arguments that 8th graders can understand if they are interested.

(d) Since 1.414213=1414213100000 , 1.414213 is a rational number.

(e) π = 3.141592 . . . is not rational. In eighth grade most students know that π is irrational as a "fact." The proof of this is quite sophisticated.

(f) Since

11 = 111 11 is rational.

(g) 17 = 0.142857 is already written in a way that makes it clear it is a rational number,

although some students might say it is irrational, possibly because the repeating part of the

decimal is longer than many familiar repeating decimals (like 13).

(h) We have

12.3456565656 = 12.34+.0056 = 1234100 +

569900 =

1234 ⋅99+569900 =

1222229900 , which is certainly

rational.

8. QUESTION: The world is consuming approximately 87 million barrels of oil per day.(a) At this rate of consumption, how long will the known world oil reserves of 1.653×1012 barrels last?

(b) Uganda has recently discovered a large deposit of oil in the Lake Albert basin. It is estimated that this deposit holds as many as 6 billion barrels of oil. In how much time would this amount be consumed by worldwide demand? Round to the nearest whole day.

ANSWER:

a) 1.9 ×104∨19,000daysb) 6.896551724 ×101∨about 70days

9. QUESTION: Convert from rational to decimal.

82100= 0.82

1312= 1.08333…

916= 0.5625

10. Order the following numbers on the number line. Make sure to put numbers on the line.

285 = 5.6 √40 ≈6.3 2√6 ≈4.9 -

12 π ≈−1.6

11. QUESTION: At 186,282 miles per second, how far does light travel in a year? Give your answer in miles, but use scientific notation, which expresses a number like 93,400,000 as 9.34×107 (which might appear on your calculator as 9.34 E7 instead). Use 365 days for a year. The answer to this question is called a light year by astronomers, who use it to measure huge distances. Other than the Sun, the star nearest the Earth is Proxima Centauri, a mere 4.2 light years away.

ANSWER: 5.874589152 x 1012

12. QUESTION: The diameter of an atom is so small that it would take about 108 of them, arranged in a line, to span one centimeter. It is thus a plausible estimate that a cubic centimeter contains about 108×108×108=¿ atoms. Write this huge number as a power of 10.

ANSWER: 1024

13. QUESTION: The average mass of an adult human is about 65 kilograms while the average mass of an ant is approximately 4 x10−3 grams. The total human population in the world is approximately 6.84 billion, and it is estimated there are currently about 10,000 trillion ants alive. 1on these values, how does the total mass of all living ants compare to the total mass of all living humans?

ANSWER: We are told the total number of ants in the world is about 10,000 trillion or 10,000 x 109 = 104 x 1012 = 1016 ants. In addition, the average mass of a single ant is 4 x 10−3 grams. Thus, the approximate total mass of all ants in the world is

(4 x 10−3g)(1016)=4 x 10−3×1016grams = 4 x 1013grams. mass for humans is given in kilograms while the mass for ants is in grams. We convert the unit of mass for a human to grams as follows,

(65kg)(103g

1kg) = (65 x 103)g = (6.5 x 101)(103)g = 6.5 x 101+3g = 6.5 x 104g.there are 6.84

billion humans on earth, the total mass of all humans on earth can be approximated as

(6.5 x 104g)(6.84 billion) = (6.5 x 104g)(6.84 x 109) = (6.5 x 6.84)(104 x 109)g = 44.46 x 104+9

g= (4.446 x 101) x 1013g = 4.446 x 1014g., the total mass of all humans in the world is greater than the total mass of all ants in the world. In fact, the calculations above show the total mass of all humans in the world is about 10 times the total mass of all ants.

3. QUESTION: Match each of the 18 measurements to one of the 9 real-life quantities.

20 m; 60 m; 0.012 m; 0.12 m; 3 m; 8000m; 400,000,000 m; 0.0001 m; 2 m

2 x 100 m; 1 x 10−¿ 4¿ m; 8 x 103 m; 3 x 100 m; 2 x 101 m; 4 x 108 m; 6 x 101 m; 1.2 x 10−¿ 1¿ m; 1.2 x 10−¿ 2 ¿ m

 

ANSWER:

 

15. QUESTION: In all these tasks you should show your calculations and give your answers to the nearest whole number.

(a) How many $3.50 burgers can you buy for a million (106) dollars?

(b) How many years does it take to earn 106 dollars if you are paid $30 an hour and work 35 hours a week for 50 weeks a year?

(c) A dollar bill weighs one gram. How many pounds do 106 dollar bills weigh? (103 grams is 1 kilogram and 1 kilogram is 2.205 pounds).

(d) A dollar bill is 0.0043 inches thick. How many yards high is a pile of 106 dollar bills?

ANSWER: (a) 285714

(b) 19-20 years

(c) 2205 pounds

(d) 119-120 yards

16. QUESTION: An ant has a mass of approximately 4 x10−3 grams and an elephant has a mass of approximately 8 metric tons.

(a) How many ants does it take to have the same mass as an elephant?

(b) An ant is 10−1 cm long. If you put all these ants from your answer to part (a) in a line (front to back), how long would the line be? Find two cities in the United States that are a similar distance apart to illustrate this length. : 1 kg = 1000 grams, 1 metric ton = 1000 kg, 1m = 100 cm, 1km = 1000 m

ANSWER:

(a) First we observe that the mass of the ant is in grams, where the mass of the elephant is in metric tons. We cannot compare the two masses as is, so we first convert them into the same units. We can do this with ratios, using the conversion chart above to convert metric tons into kilograms and then kilograms into grams.

(8 metric tons)(1000 kg1metricton )(

1000grams1kg ) = 8,000,000 grams = 8 x106 grams that both

metric tons and kg appear once on both the top and the bottom, enabling them to be canceled out, leaving only grams. Also note that we could have converted .003 grams into metric tons, and though this would have given an answer with a large number of decimal places, we would end up with the same final result. that both quantities are in grams, we want to find how many ants, n, are required to have the same mass as the elephant. As each ant has a mass of 4 x10−3 grams, we multiply 4 x10−3 by n to get the total mass of the ants mass of n ants = 4 x10−3n then we set this equal to the mass of the elephant: 4 x10−3n = 8 x106 find that n = 2 x109 ants. , it takes approximately 2 x109 ants to have the same mass as an elephant.

(b) Each ant is 10−1 cm long, and we have 2 x109 ants total. If we put all these ants in a line (front to back), we can find the length of the line by multiplying the length of one ant by how many ants we have:

2 x109 x10−1 cm = 2 x108 cm is, this answer is difficult to illustrate because it is such a large number. To find two objects that are this distance apart, we first want to convert centimeters into more useful units. We do this by using ratios from the above chart, converting centimeters to meters and meters to kilometers.

(2 x 108 cm)(1m102cm )(

1km103m) = 2 x103 km = 2000 km amount is much easier to visualize, and

it turns out that 2000 km is approximately the driving distance from San Francisco, California to Denver, Colorado.