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Basic Arithmetic
TERMINOLOGY
Absolute value: The distance of a number from zero on the number line. Hence it is the magnitude or value of a number without the sign Directed numbers: The set of integers or whole numbers f -3, -2, -1, 0, 1, 2, 3, f Exponent: Power or index of a number. For example 23 has a base number of 2 and an exponent of 3 Index: The power of a base number showing how many times this number is multiplied by itself e.g. 2 3 = 2 # 2 # 2. The index is 3 Indices: More than one index (plural) Recurring decimal: A repeating decimal that does not terminate e.g. 0.777777 … is a recurring decimal that can be written as a fraction. More than one digit can recur e.g. 0.14141414 ... Scientiﬁc notation: Sometimes called standard notation. A standard form to write very large or very small numbers as a product of a number between 1 and 10 and a power of 10 e.g. 765 000 000 is 7.65 # 10 8 in scientific notation

Chapter 1 Basic Arithmetic

3

INTRODUCTION
THIS CHAPTER GIVES A review of basic arithmetic skills, including knowing the

correct order of operations, rounding off, and working with fractions, decimals and percentages. Work on significant figures, scientific notation and indices is also included, as are the concepts of absolute values. Basic calculator skills are also covered in this chapter.

Real Numbers
Types of numbers
Unreal or imaginary numbers Real numbers

Rational numbers

Irrational numbers

Integers

Integers are whole numbers that may be positive, negative or zero. e.g. - 4, 7, 0, -11 a Rational numbers can be written in the form of a fraction b • 3 where a and b are integers, b ! 0. e.g. 1 , 3.7, 0. 5, - 5 4 a Irrational numbers cannot be written in the form of a fraction (that b is, they are not rational) e.g. 2 , r

EXAMPLE
Which of these numbers are rational and which are irrational? • 3 r 3 , 1. 3, , 9 , , - 2.65 4 5

Solution r are irrational as they cannot be written as fractions (r is irrational). 4 • 3 13 1 1. 3 = 1 , 9 = and - 2.65 = - 2 so they are all rational. 3 1 20 3 and

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Maths In Focus Mathematics Preliminary Course

Order of operations
1. Brackets: do calculations inside grouping symbols ﬁrst. (For example, a fraction line, square root sign or absolute value sign can act as a grouping symbol.) 2. Multiply or divide from left to right. 3. Add or subtract from left to right.

EXAMPLE
Evaluate 40 - 3 ] 5 + 4 g .

Solution
40 - 3 (5 + 4) = 40 - 3# 9 = 40 - 27 = 13

BRACKETS KEYS
Use ( and ) to open and close brackets. Always use them in pairs. For example, to evaluate 40 - 3 ] 5 + 4 g press 40 - 3 # ( 5 + 4 ) = = 13 5.67 - 3.49 correct to 1 decimal place To evaluate 1.69 + 2.77 press : ( ( 5.67 - 3.49 ) ' ( 1.69 + 2.77 ) ) = = 0.7 correct to 1 decimal place

PROBLEM
What is wrong with this calculation? Evaluate 19 - 4 1+2 Press 19 - 4 ' 1 + 2 = 19 - 4 '1 + 2

17

What is the correct answer?

Chapter 1 Basic Arithmetic

5

MEMORY KEYS
Use STO to store a number in memory. There are several memories that you can use at the same time—any letter from A to F, or X, Y and M on the keypad. To store the number 50 in, say, A press 50 STO A To recall this number, press ALPHA A = To clear all memories press SHIFT CLR

Different calculators use different keys so check the instructions for your calculator.

X -1 KEY
Use this key to find the reciprocal of x. For example, to evaluate 1 - 7.6 # 2.1 -1 = press ( (-) 7.6 # 2.1 ) x = - 0.063 (correct to 3 decimal places)

Rounding off
Rounding off is often done in everyday life. A quick look at a newspaper will give plenty of examples. For example in the sports section, a newspaper may report that 50 000 fans attended a football match. An accurate number is not always necessary. There may have been exactly 49 976 people at the football game, but 50 000 gives an idea of the size of the crowd.

EXAMPLES
1. Round off 24 629 to the nearest thousand.

Solution
This number is between 24 000 and 25 000, but it is closer to 25 000.

` 24 629 = 25 000 to the nearest thousand

CONTINUED

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Maths In Focus Mathematics Preliminary Course

2. Write 850 to the nearest hundred.

Solution
This number is exactly halfway between 800 and 900. When a number is halfway, we round it off to the larger number. ` 850 = 900 to the nearest hundred

In this course you will need to round off decimals, especially when using trigonometry or logarithms. To round a number off to a certain number of decimal places, look at the next digit to the right. If this digit is 5 or more, add 1 to the digit before it and drop all the other digits after it. If the digit to the right is less than 5, leave the digit before it and drop all the digits to the right.

EXAMPLES
1. Round off 0.6825371 correct to 1 decimal place.
Add 1 to the 6 as the 8 is greater than 5.

Solution
0.6825371 # ` 0.6825371 = 0.7 correct to 1 decimal place 2. Round off 0.6825371 correct to 2 decimal places.

Drop off the 2 and all digits to the right as 2 is smaller than 5.

Solution
0.6825371 # ` 0.6825371 = 0.68 correct to 2 decimal places 3. Evaluate 3.56 ' 2.1 correct to 2 decimal places.

Check this on your calculator. Add 1 to the 69 as 5 is too large to just drop off.

Solution
3.56 ' 2.1 = 1.69 #238095 5 = 1.70 correct to 2 decimal places

Chapter 1 Basic Arithmetic

7

FIX KEY
Use MODE or SET UP to fix the number of decimal places (see the instructions for your calculator). This will cause all answers to have a fixed number of decimal places until the calculator is turned off or switched back to normal.

While using a fixed number of decimal places on the display, the calculator still keeps track internally of the full number of decimal places.

EXAMPLE
Calculate 3.25 ' 1.72 # 5.97 + 7.32 correct to 2 decimal places.

Solution
3.25 ' 1.72 # 5.97 + 7.32 = 1.889534884 # 5.97 + 7.32 = 11.28052326 + 7.32 = 18.60052326 = 18.60 correct to 2 decimal places If the FIX key is set to 2 decimal places, then the display will show 2 decimal places at each step. 3.25 ' 1.72 # 5.97 + 7.32 = 1.89 # 5.97 + 7.32 = 11.28 + 7.32 = 18.60 If you then set the calculator back to normal, the display will show the full answer of 18.60052326.

Don’t round off at each step of a series of calculations.

The calculator does not round off at each step. If it did, the answer might not be as accurate. This is an important point, since some students round off each step in calculations and then wonder why they do not get the same answer as other students and the textbook.

1.1 Exercises
1. State which numbers are rational and which are irrational. (a) 169 (b) 0.546 (c) -17 r (d) 3 (e) 0.34 (f) 218 (g) 2 2 1 (h) 27 (i) 17.4% 1 (j) 5

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Maths In Focus Mathematics Preliminary Course

2.

Evaluate (a) 20 - 8 ' 4 (b) 3 # 7 - 2 # 5 (c) 4 # ] 27 ' 3 g ' 6 (d) 17 + 3 # - 2 (e) 1.9 - 2 # 3.1 14 ' 7 (f) -1 + 3 3 1 2 (g) 2 - # 5 5 3 3 1 1 4 8 (h) 5 6 5 5 ' 8 6 (i) 1 1 + 4 8 1 7 3 5 10 (j) 1 1 1 4 2 Evaluate correct to 2 decimal places. (a) 2.36 + 4.2 ' 0.3 (b) ] 2.36 + 4.2 g ' 0.3 (c) 12.7 # 3.95 ' 5.7 (d) 8.2 ' 0.4 + 4.1# 0.54 (e) ] 3.2 - 6.5 g # ] 1.3 + 2.7 g 1 (f) 4.7 + 1.3 1 (g) 4.51 + 3.28 0.9 + 1.4 (h) 5.2 - 3.6 5.33 + 2.87 (i) 1.23 - 3.15 (j) 1.7 2 + 8.9 2 - 3.94 2

7.

A crowd of 10 739 spectators attended a tennis match. Write this figure to the nearest thousand. A school has 623 students. What is this to the nearest hundred? A bank made loans to the value of \$7 635 718 last year. Round this off to the nearest million.

8.

9.

10. A company made a profit of \$34 562 991.39 last year. Write this to the nearest hundred thousand. 11. The distance between two cities is 843.72 km. What is this to the nearest kilometre? 12. Write 0.72548 correct to 2 decimal places. 13. Round off 32.569148 to the nearest unit. 14. Round off 3.24819 to 3 decimal places. 15. Evaluate 2.45 # 1.72 correct to 2 decimal places. 16. Evaluate 8.7 ' 5 correct to 1 decimal place. 17. If pies are on special at 3 for \$2.38, find the cost of each pie. 18. Evaluate 7.48 correct to 2 decimal places. 19. Evaluate 6.4 + 2.3 correct to 8 1 decimal place.

3.

4.

Round off 1289 to the nearest hundred. Write 947 to the nearest ten. Round off 3200 to the nearest thousand.

5. 6.

20. Find the length of each piece of material, to 1 decimal place, if 25 m of material is cut into 7 equal pieces.

Chapter 1 Basic Arithmetic

9

21. How much will 7.5 m 2 of tiles cost, at \$37.59 per m2? 22. Divide 12.9 grams of salt into 7 equal portions, to 1 decimal place. 23. The cost of 9 peaches is \$5.72. How much would 5 peaches cost? 24. Evaluate correct to 2 decimal places. (a) 17.3 - 4.33 # 2.16 (b) 8.72 # 5.68 - 4.9 # 3.98

(c)

3.5 + 9.8 5.6 + 4.35 15.9 + 6.3 - 7.8 (d) 7.63 - 5.12 1 (e) 6.87 - 3.21

25. Evaluate

9.91 - ] 9.68 - 5.47 g 5.39 2 correct to 1 decimal place.

DID YOU KNOW?
In building, engineering and other industries where accurate measurements are used, the number of decimal places used indicates how accurate the measurements are. For example, if a 2.431 m length of timber is cut into 8 equal parts, according to the calculator each part should be 0.303875 m. However, a machine could not cut this accurately. A length of 2.431 m shows that the measurement of the timber is only accurate to the nearest mm (2.431 m is 2431 mm). The cut pieces can also only be accurate to the nearest mm (0.304 m or 304 mm). The error in measurement is related to rounding off, as the error is half the smallest measurement. In the above example, the measurement error is half a millimetre. The length of timber could be anywhere between 2430.5 mm and 2431.5 mm.

Directed Numbers
Many students use the calculator with work on directed numbers (numbers that can be positive or negative). Directed numbers occur in algebra and other topics, where you will need to remember how to use them. A good understanding of directed numbers will make your algebra skills much better.

^ - h KEY
Use this key to enter negative numbers. For example, press (-) 3 =

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Maths In Focus Mathematics Preliminary Course

To add: move to the right along the number line To subtract: move to the left along the number line
-4 -3 -2 -1

0

2

3

4

Subtract

EXAMPLES
You can also do these on a calculator, or you may have a different way of working these out.

Evaluate 1. - 4 + 3

Solution
Start at - 4 and move 3 places to the right.

-4

-3

-2

-1

0

1

2

3

4

- 4 + 3 = -1 2. -1 - 2

Solution
Start at -1 and move 2 places to the left.

-4

-3

-2

-1

0

1

2

3

4

-1 - 2 = -3

Multiplying and dividing
To multiply or divide, follow these rules. This rule also works if there are two signs together without a number in between e.g. 2 - -3

Same signs = + + + =+ - - =+ Different signs = + - =- + =-

Chapter 1 Basic Arithmetic

11

EXAMPLES
Evaluate 1. - 2 #7

Solution
Different signs (- 2 and + 7) give a negative answer. - 2 # 7 = -14 2. -12 ' - 4

Solution
Same signs (-12 and - 4) give a positive answer. -12 ' - 4 = 3 3. -1 - - 3

Solution
The signs together are the same (both negative) so give a positive answer. -1 - 3 = -1 + 3 =2
-

Start at -1 and move 3 places to the right.

1.2 Exercises
Evaluate 1. 2. 3. 4. 5. 6. 7. 8. 9. -2 + 3 -7 - 4 8 # -7 7 - ]-3 g 28 ' -7 - 4.9 + 3.7 - 2.14 - 5.37 4.8 # -7.4 1.7 - ] - 4.87 g 3 2 -1 5 3 11. 5 - 3 # 4 12. - 2 + 7 # - 3 13. 4 - 3 # - 2 14. -1 - -2 15. 7 + - 2 16. 2 - ] -1 g 17. - 2 + 15 ' 5 18. - 2 # 6 # - 5 19. - 28 ' -7 # - 5 20. ] - 3 g2

10. -

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Maths In Focus Mathematics Preliminary Course

Fractions, Decimals and Percentages
Conversions
You can do all these conversions on your calculator using the b a or S + D key. c

EXAMPLES
1. Write 0.45 as a fraction in its simplest form.

Solution
0.45 = 45 5 ' 5 100 9 = 20 3 to a decimal. 8

3 means 3 ' 8. 8

2. Convert

Solution
0.375 8 g 3.000 3 So = 0.375 8 3. Change 35.5% to a fraction.

Solution
35.5% = 35.5 2 # 100 2 71 = 200

4. Write 0.436 as a percentage.

Solution
Multiply by 100% to change a fraction or decimal to a percentage.

0.436 = 0.436 #100% = 43.6% 5. Write 20 g as a fraction of 1 kg in its simplest form.

Solution
1 kg = 1000 g 20 g 20 g = 1000 g 1 kg 1 = 50

Chapter 1 Basic Arithmetic

13

6. Find the percentage of people who prefer to drink Lemon Fuzzy, if 24 out of every 30 people prefer it.

Solution
24 100% # = 80% 30 1

Sometimes decimals repeat, or recur. Example • 1 = 0.33333333 f = 0. 3 3 There are different methods that can be used to change a recurring decimal into a fraction. Here is one way of doing it. Later you will discover another method when studying series. (See HSC Course book, Chapter 8.)

EXAMPLES
1. Write 0. 4 as a rational number.

A rational number is any number that can be written as a fraction.

Solution
Let n = 0.44444 f Then 10n = 4.44444 f (2) - (1): 9n = 4 4 n= 9
• •

(1 ) (2 )

Check this on your calculator by dividing 4 by 9.

2. Change 1.329 to a fraction.

Solution n = 1.3292929 f Let Then 100n = 132.9292929 f (2) - (1): 99n = 131.6 131.6 10 n= # 99 10 1316 = 990 163 =1 495 (1 ) (2 )
Try multiplying n by 10. Why doesn’t this work?

CONTINUED

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Maths In Focus Mathematics Preliminary Course

Another method
This method avoids decimals in the fraction at the end.

Let n = 1.3292929 f Then 10n = 13.2929292 f and 1000n = 1329.292929 f (2) - (1): 990n = 1316 1316 n= 990 163 =1 495

(1 ) (2 )

1.3 Exercises
1. Write each decimal as a fraction in its lowest terms. (a) 0.64 (b) 0.051 (c) 5.05 (d) 11.8 Change each fraction into a decimal. 2 (a) 5 7 (b) 1 8 5 (c) 12 7 (d) 11 Convert each percentage to a fraction in its simplest form. (a) 2% (b) 37.5% (c) 0.1% (d) 109.7% Write each percentage as a decimal. (a) 27% (b) 109% (c) 0.3% (d) 6.23% Write each fraction as a percentage. 7 20 1 (b) 3 (a) (c) 2 4 15 1 (d) 1000

6.

2.

Write each decimal as a percentage. (a) 1.24 (b) 0.7 (c) 0.405 (d) 1.2794 Write each percentage as a decimal and as a fraction. (a) 52% (b) 7% (c) 16.8% (d) 109% (e) 43.4% 1 (f) 12 % 4 Write these fractions as recurring decimals. 5 (a) 6 7 (b) 99 13 (c) 99 1 (d) 6 2 (e) 3

7.

3.

8.

4.

5.

Chapter 1 Basic Arithmetic

15

5 33 1 (g) 7 2 (h) 1 11 (f) 9. Express as fractions in lowest terms. (a) 0. 8 (b) (c) (d) (e) (f) (g) 0. 2 • 1. 5 • 3. 7 • • 0. 67 • • 0. 54 • 0.15
• • •

31 99 13 + 6 (e) 7+4 (d) 1 11. Evaluate and write as a fraction. (a) 7.5 ' ] 4.1 + 7.9 g 15.7 - 8.9 (b) 4.5 - 1.3 6.3 + 1.7 (c) 12.3 - 8.9 + 7.6 4 .3 (d) 11.5 - 9.7 64 (e) 8100 12. Angel scored 17 out of 23 in a class test. What was her score as a percentage, to the nearest unit? 13. A survey showed that 31 out of 40 people watched the news on Monday night. What percentage of people watched the news? 14. What percentage of 2 kg is 350 g? 15. Write 25 minutes as a percentage of an hour.

(h) 0.216 • • (i) 0.2 19 • • (j) 1.074 10. Evaluate and express as a decimal. 5 (a) 3+6 (b) 8 - 3 ' 5 4+7 (c) 12 + 3

Investigation
Explore patterns in recurring decimals by dividing numbers by 3, 6, 9, 11, and so on. Can you predict what the recurring decimal will be if a fraction has 3 in the denominator? What about 9 in the denominator? What about 11? Can you predict what fraction certain recurring decimals will be? What denominator would 1 digit recurring give? What denominator would you have for 2 digits recurring?

Operations with fractions, decimals and percentages
You will need to know how to work with fractions without using a calculator, as they occur in other areas such as algebra, trigonometry and surds.

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Maths In Focus Mathematics Preliminary Course

The examples on fractions show how to add, subtract, multiply or divide fractions both with and without the calculator. The decimal examples will help with some simple multiplying and the percentage examples will be useful in Chapter 8 of the HSC Course book when doing compound interest. Most students use their calculators for decimal calculations. However, it is important for you to know how to operate with decimals. Sometimes the calculator can give a wrong answer if the wrong key is pressed. If you can estimate the size of the answer, you can work out if it makes sense or not. You can also save time by doing simple calculations in your head.

DID YOU KNOW?
Some countries use a comma for the decimal point—for example, 0,45 for 0.45. This is the reason that our large numbers now have spaces instead of commas between digits—for example, 15 000 rather than 15,000.

EXAMPLES
1. Evaluate 1 3 2 - . 5 4

Solution
1 3 3 2 7 - = 5 4 5 4 28 15 = 20 20 13 = 20 1 ' 3. 2

2. Evaluate 2

Solution
2 3 5 1 '3 = ' 2 2 1 5 1 = # 2 3 5 = 6

3. Evaluate 0.056 # 100.
Move the decimal point 2 places to the right.

Solution
0.056 #100 = 5.6

Chapter 1 Basic Arithmetic

17

4. Evaluate 0.02 # 0.3.

Solution
0.02 # 0.3 = 0.006 5. Evaluate 8.753 . 10

Multiply the numbers and count the number of decimal places in the question.

Solution
8.753 ' 10 = 0.8753 1 6. The price of a \$75 tennis racquet increased by 5 %. Find the new 2 price.

Move the decimal point 1 place to the left.

Solution
1 5 % = 0.055 2 1 ` 5 % of \$75 = 0.055#\$75 2 = \$4.13 1 or 105 % of \$75 = 1.055#\$75 2 = \$79.13

So the price increases by \$4.13 to \$79.13. 7. The price of a book increased by 12%. If it now costs \$18.00, what did it cost before the price rise?

Solution
The new price is 112% (old price 100%, plus 12%) \$18.00 ` 1% = 112 \$18.00 100 100% = # 112 1 = \$16.07 So the old price was \$16.07.

1.4 Exercises
1. Write 18 minutes as a fraction of 2 hours in its lowest terms. Write 350 mL as a fraction of 1 litre in its simplest form. Evaluate 3 1 (a) + 5 4 2 7 -2 5 10 3 2 (c) #1 5 4 3 (d) ' 4 7 3 2 (e) 1 ' 2 5 3 (b) 3

2.

3.

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Maths In Focus Mathematics Preliminary Course

4. 5.

Find

3 of \$912.60. 5

15. Increase 350 g by 15%. 1 16. Decrease 45 m by 8 %. 2 17. The cost of a calculator is now \$32. If it has increased by 3.5%, how much was the old cost? 18. A tree now measures 3.5 m, which is 8.3% more than its previous year’s height. How high was the tree then, to 1 decimal place?

5 Find of 1 kg, in grams correct 7 to 1 decimal place. Trinh spends sleeping, 1 of her day 3

6.

7 1 at work and 24 12

eating. What fraction of the day is left? 7. I get \$150.00 a week for a casual 1 job. If I spend on bus fares, 10 2 1 on lunches and on outings, 15 3 how much money is left over for savings? John grew by 17 of his height 200 this year. If he was 165 cm tall last year, what is his height now, to the nearest cm?

8.

9.

Evaluate (a) 8.9 + 3 (b) 9 - 3.7 (c) 1.9 #10 (d) 0.032 #100 (e) 0.7 # 5 (f) 0.8 # 0.3 (g) 0.02 # 0.009 (h) 5.72 #1000 8.74 (i) 100 (j) 3.76 # 0.1

19. This month there has been a 4.9% increase in stolen cars. If 546 cars were stolen last month, how many were stolen this month? 20. George’s computer cost \$3500. If it has depreciated by 17.2%, what is the computer worth now?

10. Find 7% of \$750. 11. Find 6.5% of 845 mL. 12. What is 12.5% of 9217 g? 13. Find 3.7% of \$289.45. 14. If Kaye makes a profit of \$5 by selling a bike for \$85, find the profit as a percentage of the selling price.

Chapter 1 Basic Arithmetic

19

PROBLEM
If both the hour hand and minute hand start at the same position at 12 o’clock, when is the first time, correct to a fraction of a minute, that the two hands will be together again?

Powers and Roots
A power (or index) of a number shows how many times a number is multiplied by itself.

EXAMPLES
1. 4 3 = 4 # 4 # 4 = 64 2. 2 5 = 2 # 2 # 2 # 2 # 2 = 32

In 4 3 the 4 is called the base number and the 3 is called the index or power.

A root of a number is the inverse of the power.

EXAMPLES
1.
3

36 = 6 since 6 2 = 36 8 = 2 since 2 3 = 8 64 = 2 since 2 6 = 64

2.

3.

6

DID YOU KNOW?
Many formulae use indices (powers and roots). For example the compound interest formula that you will study in Chapter 8 of the HSC n Course book is A = P ^ 1 + r h 4 Geometry uses formulae involving indices, such as V = rr 3. Do you know what this 3 formula is for? In Chapter 7, the formula for the distance between 2 points on a number plane is d= (x 2 - x 1) + (y 2 - y 1)
2 2

See if you can find other formulae involving indices.

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Maths In Focus Mathematics Preliminary Course

POWER AND ROOT KEYS
Use the x 2 and x 3 keys for squares and cubes. y Use the x or ^ key to find powers of numbers.

Use the Use the Use the
3 x

key for square roots. key for cube roots. for other roots.

These laws work for any m and n, including fractions and negative numbers.

Index laws
There are some general laws that simplify calculations with indices.

am # an = am + n

Proof a m # a n = (a # a #f# a) # (a # a #f# a) 14444244443 14444244443 m times n times = a444#f# a # a 2444 3 1 4 4 m + n times = am + n

am ' an = am - n

Proof am ' an = am an a # a #f# a (m times) = a # a #f# a (n times) a # a #f# a (m - n times) = 1 = am - n

(a m)n = a mn

Proof
(a m) n = a m # a m # a m #f# a m = am + m + m + f + m = a mn (n times) (n times)

Chapter 1 Basic Arithmetic

21

(ab) n = a n b n

Proof
(ab) n = ab # ab # ab #f# ab (n times) = (a # a #f# a) # (b # b #f# b) 14444244443 14444244443 n times n times = an bn

a n an c m = n b b

Proof a n a a a a c m = # # #f# b b b b b a # a # a #f # a = b # b # b #f # b an = n b (n times) (n times) (n times)

EXAMPLES
Simplify 1. m 9 # m 7 ' m 2

Solution m9 #m7 ' m2 = m9 + 7 - 2 = m 14 2. (2y 4)3

Solution
(2y 4) 3 = 2 3 (y 4) 3 = 23 y4 # 3 = 8y 12

CONTINUED

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Maths In Focus Mathematics Preliminary Course

3.

(y 6) 3 # y - 4 y5

Solution
(y 6) 3 # y - 4 y5 = = = y 18 # y - 4 y 18 + (- 4) y5 y
14

y5

y5 = y9

1.5 Exercises
1. Evaluate without using a calculator. (a) 5 3 # 2 2 (b) 3 4 + 8 2 1 3 (c) c m 4 (d) (e) 2.
3 4

(h) (i) (j) (k)

(x 7) 3 (2x 5) 2 (3y - 2) 4 a3 #a5 ' a7
5

27 16

Evaluate correct to 1 decimal place. (a) 3.7 2 (b) 1.06 1.5 (c) 2.3 - 0.2 (d) 3 19 (e) 3 34.8 - 1.2 # 43.1 1 (f) 3 0.99 + 5.61 Simplify (a) a 6 # a 9 # a 2 (b) y 3 # y - 8 # y 5 (c) a -1 # a -3 (d) w 2 # w 2 (e) x 6 ' x (f) p 3 ' p - 7 y 11 (g) 5 y
1 1

x2 p y9 w6 # w7 (m) w3 2 p #(p 3) 4 (n) p9 6 x ' x7 (o) x2 2 a # ( b 2) 6 (p) a4 # b9 (x 2) - 3 #(y 3) 2 (q) x -1 # y 4 (l) f 4. Simplify (a) x 5 # x 9 (b) a -1 # a - 6 m7 (c) m3 (d) k 13 # k 6 ' k 9 (e) a - 5 # a 4 # a - 7 (f) x 5 # x 5 m5 # n4 (g) 4 m # n2
2 3

3.

Chapter 1 Basic Arithmetic

23

(h)

p2 # p2
2

1

1

10. (a) Simplify

p5 q8 r4 p4 q6 r2

. p5 q8 r4

p (i) (3x 11) 2 (x 4) 6 (j) x3

(b) Hence evaluate

5.

Simplify (a) (pq 3) 5 a 8 (b) c m b 4a 3 (c) d 4 n b (d) (7a5b)2 m4 xy 3 #(xy 2) 4 (f) xy 8 4 (2k ) (g) (6k 3) 3 y 12 7 (h) _ 2y 5 i # 8 (i) e (j) f a6 # a4 o a 11 3 5xy 9 x8 # y3
-3

as a p4 q6 r2 7 2 fraction when p = , q = and 8 3 3 r= . 4
1 4 3

2 6 11. Evaluate (a ) when a = c m . 3 12. Evaluate b= 2 . 3 x4 y7 x5 y5 when x = 1 and 3 a3 b6 1 when a = and 2 b4

(e)

(2m 7) 3

13. Evaluate y= 2 . 9

14. Evaluate

k-5 1 when k = . 3 k-9 a4 b6 3 when a = and 3 2 2 4 a (b )

15. Evaluate b= 1 . 9

p

6.

Evaluate a3b2 when a = 2 and 3 b= . 4 If x = of 2 1 and y = , find the value 3 9 .

16. Evaluate

a6 # b3 as a fraction a5 # b2 3 1 when a = and b = . 4 9

7.

x3 y2 xy 5

17. Evaluate

8.

If a =

1 1 1 , b = and c = , 4 2 3 a2 b3 evaluate 4 as a fraction. c a b . a8 b7 a 11 b 8 when a8 b7
11 8

a2 b7 as a fraction in a3 b 2 4 index form when a = c m and 5 5 3 b=c m. 8 as a fraction a (b 2) 4 c 3 6 1 7 when a = , b = and c = . 7 3 9 (a 3) 2 b 4 c

18. Evaluate

9.

(a) Simplify

(b) Hence evaluate a=

5 2 and b = as a fraction. 5 8

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Maths In Focus Mathematics Preliminary Course

Negative and zero indices

Class Investigation
Explore zero and negative indices by looking at these questions. For example simplify x 3 ' x 5 using (i) index laws and (ii) cancelling. (i) x 3 ' x 5 = x - 2 by index laws 3 x# x# x (ii) x = 5 x# x# x# x# x x 1 = 2 x 1 So x - 2 = 2 x Now simplify these questions by (i) index laws and (ii) cancelling. (a) x 2 ' x 3 (b) x 2 ' x 4 (c) x 2 ' x 5 (d) x 3 ' x 6 (e) x 3 ' x 3 (f) x 2 ' x 2 (g) x ' x 2 (h) x 5 ' x 6 (i) x 4 ' x 7 (j) x ' x 3 Use your results to complete: x0 = x-n =

x0 = 1

Proof xn ' xn = xn - n = x0 xn xn ' xn = n x =1 ` x0 = 1

Chapter 1 Basic Arithmetic

25

x-n =

1 xn

Proof x0 ' xn = x0 - n = x-n x0 x0 ' xn = n x 1 = n x 1 ` x-n = n x

EXAMPLES
1. Simplify e ab 5 c o . abc 4
0

Solution e ab 5 c o =1 abc 4

0

2. Evaluate 2 - 3 .

Solution
2-3 = 1 23 1 = 8

3. Write in index form. 1 x2 3 (b) 5 x 1 (c) 5x 1 (d) x +1 (a)
CONTINUED

26

Maths In Focus Mathematics Preliminary Course

Solution
1 = x-2 x2 3 (b) 5 = 3# 15 x x -5 = 3x 1 1 1 = #x (c) 5x 5 1 -1 = x 5 1 1 = (d) x +1 (x + 1) 1 = ] x + 1 g-1 (a) 4. Write a−3 without the negative index.

Solution a-3 = 1 a3

1.6 Exercises
1. Evaluate as a fraction or whole number. (a) 3 - 3 (b) 4 - 1 (c) 7 - 3 (d) 10 - 4 (e) 2 - 8 (f) 60 (g) 2 - 5 (h) 3 - 4 (i) 7 - 1 (j) 9 - 2 (k) 2 - 6 (l) 3 - 2 (m) 40 (n) 6 - 2 (o) 5 - 3 (p) 10 - 5 (q) 2 - 7 (r) 2 0 (s) 8 - 2 (t) 4 - 3 2. Evaluate (a) 2 0 1 -4 (b) c m 2 2 -1 (c) c m 3 5 -2 (d) c m 6 x + 2y 0 p (e) f 3x - y 1 -3 (f) c m 5 3 -1 (g) c m 4 1 -2 (h) c m 7 2 -3 (i) c m 3 1 -5 (j) c m 2 3 -1 (k) c m 7

Chapter 1 Basic Arithmetic

27

8 0 (l) c m 9 6 -2 (m)c m 7 9 -2 (n) c m 10 6 0 (o) c m 11 1 -2 (p) c - m 4 2 -3 (q) c - m 5 2 -1 (r) c - 3 m 7 3 0 (s) c - m 8 1 -2 (t) c - 1 m 4 3. Change into index form. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) 1 m3 1 x 1 p7 1 d9 1 k5 1 x2 2 x4 3 y2 1 2z 6 3 5t 8 2 7x

1 (3x + 4) 2 1 (o) ( a + b) 8 1 (p) x-2 (n) 1 (5p + 1) 3 2 (r) (4t - 9) 5 1 (s) 4 (x + 1) 11 5 (t) 9 ( a + 3 b) 7 (q) 4. Write without negative indices. (a) t - 5 (b) x - 6 (c) y - 3 (d) n - 8 (e) w - 10 (f) 2x -1 (g) 3m - 4 (h) 5x - 7 (i) ]2xg- 3

(j) ] 4n g-1 (k) ] x + 1 g- 6

(n) ^ 3x + 2y h- 9 1 -5 (o) b x l 1 -10 (p) c y m 2 -1 (q) d n p 1 -2 m a+b x + y -1 (s) e x - y o (r) c (t) e 2w - z - 7 o 3x + y

(l) ^ 8y + z h-1 (m) ]k - 3g- 2

5 2m 6 2 (m) 7 3y

28

Maths In Focus Mathematics Preliminary Course

Fractional indices

Class Investigation
Explore fractional indices by looking at these questions. For example simplify (i) ` x 2 j and (ii) ^ x h .
1 2 2

(i) ` x 2 j = x 1 =x 2 (ii) ^ x h = x
1 2 1

1 2

^ by index laws h

So ` x 2 j = ^ x h = x
2

`

x2 =
1

x

Now simplify these questions. (a) ^ x 2 h 2 (b) x2
1 3 1

(c) ` x 3 j

(d) ^ x 3 h 3
3

3 (e) ^ 3 x h

(f)

x3
1 4 1

(h) ^ x 4 h 4
4

4 (g) ` x j

4 (i) ^ 4 x h

(j)

x4

Use your results to complete: xn =
1

a =n a

1 n

Proof
`an j = a ^ n a hn = a
1 n

^ by index laws h

` a =n a

1 n

Chapter 1 Basic Arithmetic

29

EXAMPLES
1. Evaluate (a) 49
1 2 1

(b) 27 3

Solution
(a) 49 = 49 =7 (b) 27 = 3 27 =3 2. Write 3x - 2 in index form.
1 3 1 2

Solution
3x - 2 = (3 x - 2) 2 3. Write (a + b) 7 without fractional indices.
1 1

Solution
( a + b) 7 = 7 a + b
1

Putting the fractional and negative indices together gives this rule.

a

1 -n

=

n

1 a

Here are some further rules.

a = n am = (n a ) m

m n

Proof n n a = `a j m = ^n a h m 1 m

a = ^ am h = n am

m n

1 n

30

Maths In Focus Mathematics Preliminary Course

a -n b n c m = bal b

Proof a -n 1 c m = b a n c m b 1 = n a bn an bn bn =1# n a bn = n a b n = bal =1'

EXAMPLES
1. Evaluate
4

(a) 8 3 (b) 125
1 3

2 -3 (c) c m 3

Solution
(a) 8 3 = (3 8 ) 4 (or 3 8 4 ) = 24 = 16 (b) 125
1 3 4

=

1
1

125 3 1 =3 125 1 = 5

Chapter 1 Basic Arithmetic

31

(c) c 2 m 3

-3

3 3 =c m 2 27 = 8 3 =3 8

2. Write in index form. (a) (b)
3

x5 1 (4x - 1) 2
2

Solution
(a) (b)
3

x5 = x 2 1 (4x - 1)
2 2

5

=

1 (4x 2 - 1) 3
2 2 3

= (4x 2 - 1)
3 5

3. Write r

without the negative and fractional indices.

Solution r 3 5

= =

1
3

r5 1
5

r3

DID YOU KNOW?
Nicole Oresme (1323–82) was the first mathematician to use fractional indices. John Wallis (1616–1703) was the first person to explain the significance of zero, negative and fractional indices. He also introduced the symbol 3 for infinity. Do an Internet search on these mathematicians and find out more about their work and backgrounds. You could use keywords such as indices and infinity as well as their names to find this information.

32

Maths In Focus Mathematics Preliminary Course

1.7 Exercises
1. Evaluate (a) 81
1 2 1

3.

Write without fractional indices.
1

(a) y 3
2

(b) 27 3
1

(b) y 3 (c) x
1 2 1

(c) 16 2
1

(d) 8 3
1

(d) (2x + 5) 2 (e) (3x - 1)
1 1 1 2

(e) 49 2 (f) 1000 3
1

(f) (6q + r) 3 (g) (x + 7) 4.
2 5

(g) 16 4
1

(h) 64 2 (i) 64 (j) 1
1 7 1 4 1 5 1 3

Write in index form. (a) (b) (c) (d) (e) (f)
3 5

t y x3 9-x 4s + 1 1 2t + 3 1 (5x - y) 3 (3x + 1) 5 1
3

(k) 81 (l) 32
1

(m) 0 8 (n) 125
1 3 1

(g) (h) (i)

(o) 343 3
1

(p) 128 7
1

(q) 256 4 (r) 9 (s) 8
3 2 1 3 2 3

(t) 64 2.

Evaluate correct to 2 decimal places.
1

(a) 23 4 (b) 4 45.8 (c) (d) (e) (f)
7

(x - 2) 2 1 (j) 2 y+7 5 (k) 3 x+4 2 (l) 3 y2 - 1 3 (m) 5 4 (x 2 + 2) 3 Write in index form and simplify. (a) x x x (b) x x (c) 3 x x2 (d) 3 x (e) x 4 x

5

8

1.24 + 4.3 2 1 12.9 3 .6 - 1 .4 1 .5 + 3 .7

5.

4

5 .9 # 3 .7 8.79 - 1.4

Chapter 1 Basic Arithmetic

33

6.

Expand and simplify, and write in index form. (a) ( x + x) 2 (b) (3 a + 3 b ) (3 a - 3 b ) 1 2 (c) f p + p p (d) ( x + (e) 1 2 ) x x ( x 2 - 3x + 1 ) x3

7.

Write without fractional or negative indices. (a) (a - 2b) (b) (y - 3)
2 3 4 7 1 3

(c) 4 (6a + 1) ( x + y) (d) 3
5 4

-

6 (3 x + 8 ) (e) 7

-

2 9

Scientiﬁc notation (standard form)
Very large or very small numbers are usually written in scientiﬁc notation to make them easier to read. What could be done to make the ﬁgures in the box below easier to read?

DID YOU KNOW?
The Bay of Fundy, Canada, has the largest tidal changes in the world. About 100 000 000 000 tons of water are moved with each tide change. The dinosaurs dwelt on Earth for 185 000 000 years until they died out 65 000 000 years ago. The width of one plant cell is about 0.000 06 m. In 2005, the total storage capacity of dams in Australia was 83 853 000 000 000 litres and households in Australia used 2 108 000 000 000 litres of water.

A number in scientiﬁc notation is written as a number between 1 and 10 multiplied by a power of 10.

EXAMPLES
1. Write 320 000 000 in scientiﬁc notation.

Solution
320 000 000 = 3.2 #10 8 2. Write 7.1#10 -5 as a decimal number.

Write the number between 1 and 10 and count the decimal places moved.

Solution
7.1#10
-5

= 7.1 ' 10 = 0.000 071
5

Count 5 places to the left.

34

Maths In Focus Mathematics Preliminary Course

SCIENTIFIC NOTATION KEY
Use the EXP or #10 x key to put numbers in scientific notation. For example, to evaluate 3.1#10 4 ' 2.5 #10 - 2, press 3.1 EXP 4 ' 2.5 EXP (-) 2 = = 1 240 000

DID YOU KNOW?
Engineering notation is similar to scientific notation, except the powers of 10 are always multiples of 3. For example, 3.5 # 10
3 -6

15.4 # 10

SIGNIFICANT FIGURES The concept of signiﬁcant ﬁgures is related to rounding off. When we look at very large (or very small) numbers, some of the smaller digits are not signiﬁcant. For example, in a football crowd of 49 976, the 6 people are not really signiﬁcant in terms of a crowd of about 50 000! Even the 76 people are not signiﬁcant. When a company makes a proﬁt of \$5 012 342.87, the amount of 87 cents is not exactly a signiﬁcant sum! Nor is the sum of \$342.87. To round off to a certain number of signiﬁcant ﬁgures, we count from the ﬁrst non-zero digit. In any number, non-zero digits are always signiﬁcant. Zeros are not signiﬁcant, except between two non-zero digits or at the end of a decimal number. Even though zeros may not be signiﬁcant, they are still necessary. For example 31, 310, 3100, 31 000 and 310 000 all have 2 signiﬁcant ﬁgures but are very different numbers! Scientiﬁc notation uses the signiﬁcant ﬁgures in a number.

EXAMPLES
12 000 = 1.2 #10 4 0.000 043 5 = 4.35#10 - 5 0.020 7 = 2.07 #10 - 2 (2 significant figures) (3 significant figures) (3 significant figures)

When rounding off to signiﬁcant ﬁgures, use the usual rules for rounding off.

Chapter 1 Basic Arithmetic

35

EXAMPLES
1. Round off 4 592 170 to 3 significant figures.

Solution
4 592 170 = 4 590 000 to 3 significant figures 2. Round off 0.248 391 to 2 significant figures.
Remember to put the 0’s in!

Solution
0.248 391 = 0.25 to 2 significant figures 3. Round off 1.396 794 to 3 significant figures.

Solution
1.396 794 = 1.40 to 3 significant figures

1.8 Exercises
1. Write in scientific notation. (a) 3 800 (b) 1 230 000 (c) 61 900 (d) 12 000 000 (e) 8 670 000 000 (f) 416 000 (g) 900 (h) 13 760 (i) 20 000 000 (j) 80 000 Write in scientific notation. (a) 0.057 (b) 0.000 055 (c) 0.004 (d) 0.000 62 (e) 0.000 002 (f) 0.000 000 08 (g) 0.000 007 6 (h) 0.23 (i) 0.008 5 (j) 0.000 000 000 07 3. Write as a decimal number. (a) 3.6 #10 4 (b) 2.78 #10 7 (c) 9.25#10 3 (d) 6.33#10 6 (e) 4 #10 5 (f) 7.23#10 - 2 (g) 9.7 #10 - 5 (h) 3.8 # 10 - 8 (i) 7 #10 - 6 (j) 5#10 - 4 Round these numbers to 2 significant figures. (a) 235 980 (b) 9 234 605 (c) 10 742 (d) 0.364 258 (e) 1.293 542 (f) 8.973 498 011 (g) 15.694 (h) 322.78 (i) 2904.686 (j) 9.0741

2.

4.

36

Maths In Focus Mathematics Preliminary Course

5.

Evaluate correct to 3 significant figures. (a) 14.6 # 0.453 (b) 4.8 ' 7 (c) 4.47 + 2.59 #1.46 1 (d) 3.47 - 2.7

6.

Evaluate 4.5#10 4 # 2.9 #10 5, giving your answer in scientific notation. 8.72 #10 - 3 and write 1.34 #10 7 your answer in standard form correct to 3 significant figures. Calculate

7.

Investigation
A logarithm is an index. It is a way of finding the power (or index) to which a base number is raised. For example, when solving 3 x = 9, the solution is x = 2. The 3 is called the base number and the x is the index or power. You will learn about logarithms in the HSC course.
The a is called the base number and the x is the index or power.

If a x = y then log a y = x

1. The expression log7 49 means the power of 7 that gives 49. The solution is 2 since 7 2 = 49. 2. The expression log2 16 means the power of 2 that gives 16. The solution is 4 since 2 4 = 16. Can you evaluate these logarithms? 1. log3 27 2. log5 25 3. log10 10 000 4. log2 64 5. log4 4 6. log7 7 7. log3 1 8. log4 2 1 9. log 3 3 1 10. log 2 4

Chapter 1 Basic Arithmetic

37

Absolute Value
Negative numbers are used in maths and science, to show opposite directions. For example, temperatures can be positive or negative.

But sometimes it is not appropriate to use negative numbers. For example, solving c 2 = 9 gives two solutions, c = !3. However when solving c 2 = 9, using Pythagoras’ theorem, we only use the positive answer, c = 3, as this gives the length of the side of a triangle. The negative answer doesn’t make sense. We don’t use negative numbers in other situations, such as speed. In science we would talk about a vehicle travelling at –60k/h going in a negative direction, but we would not commonly use this when talking about the speed of our cars!

Absolute value definitions
We write the absolute value of x as x
We can also define x as the distance of x from 0 on the number line. We will use this in Chapter 3.

x =)

x when x \$ 0 - x when x 1 0

EXAMPLES
1. Evaluate 4 .

Solution
4 = 4 since 4 \$ 0

CONTINUED

38

Maths In Focus Mathematics Preliminary Course

2. Evaluate - 3 .

Solution
-3 = - ] - 3 g since - 3 1 0 =3

The absolute value has some properties shown below.

Properties of absolute value

| ab | = | a |#| b | |a | = a
2 2

e.g. | 2 # - 3 | = | 2 |#| - 3 | = 6 e.g. | - 3 | 2 = ] - 3 g2 = 9 e.g. 5 2 = | 5 | = 5 e.g. | -7 | = | 7 | = 7 e.g. | 2 - 3 | = | 3 - 2 | = 1 e.g. | 2 + 3 | = | 2 | + | 3 | but | - 3 + 4 | 1 | - 3 | + | 4 |

a2 = | a | |- a | = | a | |a - b | = | b - a | | a + b |#| a | + | b |

EXAMPLES
1. Evaluate 2 - -1 + - 3 2.

Solution
2 - -1 + - 3 2 = 2 - 1 + 3 2 =2 -1 + 9 = 10 2. Show that a + b # a + b when a = - 2 and b = 3.

Solution
LHS means Left Hand Side.

LHS = a + b = -2 + 3 = 1 =1

Chapter 1 Basic Arithmetic

39

RHS = a + b = -2 + 3 = 2+3 =5 Since 11 5 a+b # a + b 3. Write expressions for 2x - 4 without the absolute value signs.

RHS means Right Hand Side.

Solution
2x - 4 = 2x - 4 when 2x - 4 \$ 0 i.e. 2x \$ 4 x\$2 2x - 4 = - ] 2x - 4 g when 2x - 4 1 0 = - 2x + 4 i.e. 2x 1 4 x12

Class Discussion
Are these statements true? If so, are there some values for which the expression is undefined (values of x or y that the expression cannot have)? 1. 2. 3. 4. 5. 6. 7. 8. x =1 x 2x = 2x 2x = 2 x x + y = x+y
2 x = x2 3 x = x3 x +1 = x +1

3x - 2 =1 3x - 2 x 9. =1 x2 10. x \$ 0 Discuss absolute value and its definition in relation to these statements.

40

Maths In Focus Mathematics Preliminary Course

1.9 Exercises
1. Evaluate (a) 7 (b) - 5 (c) - 6 (d) 0 (e) 2 (f) -11 (g) - 2 3 (h) 3 - 8 2 (i) - 5 (j) - 5 3 Evaluate (a) 3 + - 2 (b) - 3 - 4 (c) - 5 + 3 (d) 2 #-7 (e) - 3 + -1 2 (f) 5 - - 2 # 6 (g) - 2 + 5# -1 (h) 3 - 4 (i) 2 - 3 - 3 - 4 (j) 5 - 7 + 4 - 2 Evaluate a - b if (a) (b) (c) (d) (e) 4. a = 5 and b = 2 a = -1 and b = 2 a = - 2 and b = - 3 a = 4 and b = 7 a = -1 and b = - 2. (h) a + 1 when a 1 -1 (i) (j) 5. x - 2 when x 2 2 x - 2 when x 1 2.

Show that a + b # a + b when (a) a = 2 and b = 4 (b) a = -1 and b = - 2 (c) a = - 2 and b = 3 (d) a = - 4 and b = 5 (e) a = -7 and b = - 3. Show that x 2 = x when (a) x = 5 (b) x = - 2 (c) x = - 3 (d) x = 4 (e) x = - 9. Use the definition of absolute value to write each expression without the absolute value signs (a) x + 5 (b) b - 3 (c) a + 4 (d) 2y - 6 (e) 3x + 9 (f) 4 - x (g) 2k + 1 (h) 5x - 2 (i) a + b (j) p - q Find values of x for which x = 3. n Simplify n where n ! 0. x-2 and state which x-2 value x cannot be.

2.

6.

7.

3.

Write an expression for (a) a when a 2 0 (b) (c) (d) (e) (f) (g) a when a 1 0 a when a = 0 3a when a 2 0 3a when a 1 0 3a when a = 0 a + 1 when a 2 -1

8. 9.

10. Simplify

Chapter 1 Basic Arithmetic

41

Test Yourself 1
1. Convert (a) 0.45 to a fraction (b) 14% to a decimal 5 (c) to a decimal 8 (d) 78.5% to a fraction (e) 0.012 to a percentage 11 (f) to a percentage 15 Evaluate as a fraction. (a) 7 - 2 (b) 5 -1 (c) 9 3.
1 2

6.

Evaluate (a) 1 (b) (c) (d) (e) 3 7 5 8 6 2 #3 7 3 3 9' 4 2 1 +2 5 10 5 15# 6

2.

7.

Evaluate (a) - 4
1

Evaluate correct to 3 significant figures. (a) 4.5 2 + 7.6 2 (b) 4.3 0.3 2 (c) 3 5.7 (d) 1.3#10 9 3.8 #10 6
2 3

(b) 36 2 (c) - 5 2 - 2 3 (d) 4 - 3 as fraction
2

(e) 8 3 (f) - 2 - 1 (g) 49
1 4 1 2

as a fraction

(e) 6 4.

Evaluate (a) |-3 | -| 2 | (b) | 4 - 5 | (c) 7 + 4 # 8 (d) [(3 + 2)#(5 - 1) - 4] ' 8 (e) - 4 + 3 - 9 (f) - 2 - -1 (g) - 24 ' - 6 Simplify (a) x 5 # x 7 ' x 3 (b) (5y 3) 2 (a 5) 4 b 7 (c) a9 b 3 2x 6 n (d) d 3 (e) e ab 4 o a5 b6
0

(h) 16 (i) ] -3 g0 (j) 4 - 7 2 - -2 - 3 8. Simplify (a) a 14 ' a 9 6 (b) _ x 5 y 3 i (c) p 6 # p 5 ' p 2

5.

4 (d) ^ 2b 9h (2x 7) 3 y 2 (e) x 10 y

9.

Write in index form. n 1 (b) 5 x 1 (c) x+y (d)
4

(a)

x +1

42

Maths In Focus Mathematics Preliminary Course

(e)

7

a+b

2 (f) x 1 (g) 2x 3 (h) (i) (j)
3 7

(c) If he spends 3 hours watching TV, what fraction of the day is this? (d) What percentage of the day does he spend sleeping? 16. The price of a car increased by 12%. If the car cost \$34 500 previously, what is its new price? 17. Rachel scored 56 out of 80 for a maths test. What percentage did she score? 18. Evaluate 2118, and write your answer in scientific notation correct to 1 decimal place. 19. Write in index form. (a) x 1 (b) y (c) x+3 1 (d) (2x - 3) 11
6

x4 (5x + 3) 9 1 m3

4

10. Write without fractional or negative indices. (a) a - 5
1

(b) n 4

(c) (x + 1) 2 (d) (x - y) -1 (e) (4t - 7) - 4 (f) (a + b) 5 (g) x
1 3 3 1

1

(h) b 4 (i) (2x + 3) (j) x
3 2 4 3

(e)

3

y7

20. Write in scientific notation. (a) 0.000 013 (b) 123 000 000 000 21. Convert to a fraction. • (a) 0. 7 • • (b) 0.124 22. Write without the negative index. (a) x - 3 (b) (2a + 5)- 1 a -5 (c) c m b 23. The number of people attending a football match increased by 4% from last week. If there were 15 080 people at the match this week, how many attended last week? 24. Show that | a + b | # a + b when a = - 2 and b = - 5.

11. Show that a + b # a + b when a = 5 and b = - 3. 9 2 12. Evaluate a b when a = and b = 1 . 25 3
2 4

3 1 4 13. If a = c m and b = , evaluate ab 3 as a 4 3 fraction. 14. Increase 650 mL by 6%. 15. Johan spends 1 of his 24-hour day 3 1 sleeping and at work. 4 (a) How many hours does Johan spend at work? (b) What fraction of his day is spent at work or sleeping?

Chapter 1 Basic Arithmetic

43

Challenge Exercise 1
1. 2. 3. Simplify c 8 3 2 2 7 + 3 m ' c4 - 1 m. 4 5 3 8 11. Show that 2 (2 k - 1) + 2 k + 1 = 2 (2 k + 1 - 1) . 12. Find the value of a in index form if b3 c2

3 5 149 7 Simplify + + . 5 12 180 30 Arrange in increasing order of size: 51%, • 51 0.502, 0. 5, . 99 1 1 of his day sleeping, 3 12 1 of the day eating and of the day 20 watching TV. What percentage of the day is left? Mark spends Write 64
2 3

3 2 2 4 1 3 a = c m , b = c - m and c = c m . 5 5 3 13. Which of the following are rational • 3 numbers: 3 , - 0.34, 2, 3r, 1. 5, 0, ? 7 14. The percentage of salt in 1 L of water is 10%. If 500 mL of water is added to this mixture, what percentage of salt is there now? 15. Simplify |x + 1 | x2 - 1 for x ! !1.

4.

5. 6.

as a rational number.
25

Express 3.2 ' 0.014 in scientific notation correct to 3 significant figures. Vinh scored 17 1 out of 20 for a maths 2 1 test, 19 out of 23 for English and 55 2 out of 70 for physics. Find his average score as a percentage, to the nearest whole percentage. Write 1.3274 as a rational number. The distance from the Earth to the moon is 3.84 #10 5 km. How long would it take a rocket travelling at 2.13#10 4 km h to reach the moon, to the nearest hour? 8.3# 4.1 correct to 0.2 + 5.4 ' 1.3 3 significant figures.
• • •

16. Evaluate 6

7.

4.3 1.3 - 2.9 correct to 2.4 3 + 3.31 2 2 decimal places.

17. Write 15 g as a percentage of 2.5 kg. 18. Evaluate 2.3 1.8 + 5.7 #10 - 2 correct to 3 significant figures. 19. Evaluate - 3.4 #10 - 3 + 1.7 #10 - 2 and (6.9 #10 5) 3 express your answer in scientific notation correct to 3 significant figures.

8. 9.

20. Prove | a + b | # | a | + | b | for all real a, b.

10. Evaluate 3

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...Vanessa Zsorey Dr. McCort College Comp 11 December 2015 Sonic Highways A band lasting as long as twenty years in the music industry with their original sound not suffering is a rarity. The Foo Fighters, led by the legendary Dave Grohl, are magnificent rockers who throughout two decades, continue to stay relevant in many ways. Throughout these twenty years they have released nine albums, each of which contained some of the biggest alternative rock hits of all time. Though they may not be riding the radio waves constantly or may only appeal to a shrinking audience, their craft has never once suffered. Their newest album, Sonic Highways, is pure proof that the Foo Fighters continuously try to push themselves out of their comfort zones in order to remain interesting to their fans. Sonic Highways was the most recent project the Foo Fighters have put out and the most creative of the past eight albums. The process to create this album was far more in depth and unique compared to the traditional methods. Typically, when a band decides they want to record, they find the closest studio that is in their price range and is the best quality necessary to fit the project. Mixing the recordings has gotten incredibly easy in recent times, where all that is really necessary is the ProTools program. With Protools, any errors can easily be cleaned and corrected, then moved on to the master CD and sent off to be pressed. A decade ago, recording and mixing were not nearly as simple or......

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#### Responsibility

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#### Cig Rolling

...Joe Polce English 015 Mrs. Wickman How To Roll Your Own Cigarette Is anyone in your life a cigarette smoker? Do you often notice that they are always complaining about their ever-rising price? According to the U.S Census Bureau, out of the 6.7 billion people in the world, 2 billion are smokers of cigarettes or some other form of tobacco. Almost 75% of these 2 billion spend between \$1000 - \$1500 yearly on their favorite packs and cartons that they just cannot resist. We know its a habit, right? Its also an expensive habit. I am going to show you a quick and easy way to not only cut the cost of cigarettes in half, but also save you hundred of dollars. A select few smokers of the modern world have been fortunate enough to find the solution to the climbing prices of cigarettes. This solution is actually very simple. Why don’t we, instead of spending the \$6 plus on an already produced pack of Marlboro Lights, make and produce our own pack of very similar cigarettes? Herein lies the solution; roll your own cigarettes. Most people shy away from such an idea claiming that they would never know where to start, or have no idea where they would buy the equipment, but on contrary the first steps to this new lifestyle are very easy, and the tools needed are readily available. The benefits of rolling your own cigarettes are numerous. First off, its inexpensive. GS Hammack, a man who writes for a website called AssociatedContact.com, said that “smokers that are looking......

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#### John Mayer

...music learned how to play several instruments by the inspiration of others. Mayer loved music, especially blues as can be heard from all his songs in which they all share a melancholic tune. As a young man in high school one of his first gigs was at a local blues bar with a band he had been currently been a part of. Although he only played guitar at the time Mayer was still songwriting. He learned how to write after being hospitalized at the age of 17 with a mental issue even though his illness was destroying his body his creativity kept flowing allowing him to find his newfound passion, songwriting. All these action leading to his first debut of his album in 2001 “Room for Squares” in which he would receive Grammy nominations for the song “Your body is a wonderland” and the public eye would be drawn towards Mayer. After this Mayer would release many albums and singles that would lead to the fame he has today. In my opinion, John Mayer’s music is one of a kind and truly beautiful all holding a deep message leaving for the listener to interpret in however which way they feel. Furthermore, out of all of Mayer’s accomplishments “Waiting on the world to change” from the album “Continuum” is one of his best works. This song has a slow tempo like most of his songs on the “Continuum” album. Also, in this song the beats are...

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#### Anti-Money Laundering Regulations and Its Impact on the Financial Service Industry

...difficulties for the banking sector. Consequently several big institutions such as HSBC and Citigroup have already been fined for failing to comply with AML laws. Regulations have changed the financial sector. This essay will analyse the impact of the 9/11 attacks on the regulatory framework as well as the effects of AML on the financial industry. ￼Page 3 of 16 2. MEASURES TAKEN AML has been on the agenda of regulators well before 9/11. However, the actions of prosecutors were more focused on fighting organised crime, drug and weapon dealing as well as tax evasion. 9/11 shifted the attention towards fighting terrorism and initiated Counter Terrorist Financing (“CTF”) laws. It was not until 9/11, that strict reporting rules known as “Know Your Customer” were introduced in the US. Before, the banking lobby could prevent further regulations, however, the US PATRIOT Act 2001, in response to 9/11, took AML and CTF measures to a new level. Since 2001 a comprehensive due diligence for high-risk products and especially for so-called “Politically Exposed Persons” is mandatory in the US....

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#### Legal Scenario

...Antonio Raimundo Montana (“Tony”) was once the office manager of Captain Kirk, P.A. (“Firm”). A few of Tony’s duties involved client billing and managing the Firm’s accounts, both the operating and trust accounts. Tony had been at the Firm for over 20 years and was a trusted employee, so much so that the managing partner, Spock, rarely supervised Tony’s actions. When Tony first started at the Firm he would provide Spock a monthly report that detailed the activity of the trust account. Spock required this report as means to comply with the Florida Bar Rule 5-1.1, which mentions that a lawyer must maintain and supervise the trust account. However, as the Firm rapidly expanded and the workload increased, Tony asked Spock to eliminate such “busy” work like that of the monthly report of the trust account. Spock agreed to cease the monthly reports since he had developed a strong sense of trust and loyalty to Tony over the years. Also, Tony had never done anything in the past to make Spock suspicious of Tony’s actions. Spock had to go out of state for a month’s time for a trial. Before leaving, he asked that Tony be on the lookout for a \$1 million dollar wire transfer that was expected to be coming in the next couple of days and is to be held within the Firm’s trust account. Tony obliged and said that everything would be taken care of and that the accounting records would be updated once received. While Spock was out of town, he emailed Tony on several occasions......

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#### The Rental Heart

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#### Gm 591

...1 Week 3 Case Study The Forgotten Group Member Janail Richard-Brown 8565 W Sam Houston PKWY S Houston, Texas 770?? Jrich_26@yahoo.com 713-988-1038 GM 591 Leadership Organizational Behavior Kim Nugent January 23, 2012 Group Development As of right now, the group is somewhere between norming and performing-this stage is actually called storming. My reason for using the two stages is that one group member is still not performing up to standard (Mike). I find that this has occurred and been encountered quite often in my experience at DeVry. Most members will be onboard and one will just do the lease amount of work necessary to pacify other members. Christine having and understanding of the stages of group development would better suit her in two ways. First she would be aware of how the group is functioning and the dynamics of a group activity. Then secondly, she would see that Mike was not falling under any of the developmental practices in the stages. Also, I think it would give her a better idea on how to obtain Mike’s full efforts on the assignment. Problem Identification Christine is first facing the lack of participation by one of the group’s members. Secondly the group is at a critical point because the assignment is nearing a close. Christine needs to understand how to motivate each member to be the best at giving a total effort to the assignment. People are a very delicate lot. She needs to have each member perform at a high standard and at the same......

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#### Dear Alina

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#### Kyc Guide

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#### Anti Money Laundering

...Anti Money Laundering http://www.antimoneylaundering.org/ HOME | ABOUT US | INDUSTRY | SERVICES | NEWS & MEDIA | EVENTS | CONTACT US Related Areas Know Your Customer (KYC) Services Online Political Corruption Data-File Terrorist Financing Filter Anti Corruption Enhanced Due Diligence (EDD) services Regulatory Compliance Country-Check Foreign Corrupt Practises Act Anti Money Laundering The word money laundering refers to the use of the financial system to hide the source of funds gained from illegal activity such as drug trafficking, bribery, extortion, embezzlement, theft or other criminal activity, as the criminals try to make their ill gotten gains appear genuine. Anti Money Laundering is the term used by banks and other financial institutions to describe the variety of measures they have to combat this illegal activity and to prevent criminals from using individual banks and the financial system in general as the conduit for their Proceeds of Crime. In all major jurisdictions around the world, criminal legislation and regulation make it mandatory for banks and financial institutions to have arrangements to combat Money Laundering, with harsh criminal penalties for non-compliance. The vast majority of criminal dealings are done in cash. Criminals need ways to dispose of the cash and have it reappear as part of their wealth with as little chance as possible of it being tracked back to the cash element. Criminals have to use the......

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