Maths in Focus - Margaret Grove - ch1

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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

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? 19 - 4 1+2 Press 19 - 4 ' 1 + 2 = 19 - 4 '1 + 2

Evaluate

What is the correct answer?

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Chapter 1 Basic Arithmetic

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

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

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

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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 # 5238095

= 1.70 correct to 2 decimal places

Chapter 1 Basic Arithmetic

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.

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

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

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Maths In Focus Mathematics Preliminary Course

2.

3.

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

7.

A crowd of 10 739 spectators attended a tennis match. Write this figure to the nearest thousand.

8.

A school has 623 students. What is this to the nearest hundred?

9.

A bank made loans to the value of $7 635 718 last year. Round this off to the nearest million.

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

13. Round off 32.569148 to the nearest unit.

0.9 + 1.4 (h) 5.2 - 3.6 5.33 + 2.87 (i) 1.23 - 3.15 (j) 4.

1.7 2 + 8.9 2 - 3.94 2

Round off 1289 to the nearest hundred.

5.

Write 947 to the nearest ten.

6.

Round off 3200 to the nearest thousand.

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.

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. 6.4 + 2.3 correct to 8 1 decimal place.

19. Evaluate

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

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

21. How much will 7.5 m 2 of tiles cost, at $37.59 per m2?

(c)

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?

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

25. Evaluate

24. Evaluate correct to 2 decimal places. (a) 17.3 - 4.33 # 2.16 (b) 8.72 # 5.68 - 4.9 # 3.98

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

Adding and subtracting

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

1

Subtract

2

3

4

Add

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

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

-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

1.2 Exercises Evaluate 1.

-2 + 3

11. 5 - 3 # 4

2.

-7 - 4

12. - 2 + 7 # - 3

3.

8 # -7

13. 4 - 3 # - 2

4.

7 - ]-3 g

14. -1 - -2

5.

28 ' -7

15. 7 + - 2

6.

- 4.9 + 3.7

16. 2 - ] -1 g

7.

- 2.14 - 5.37

17. - 2 + 15 ' 5

8.

4.8 # -7.4

18. - 2 # 6 # - 5

9.

1.7 - ] - 4.87 g

19. - 28 ' -7 # - 5

10. -

3 2 -1 5 3

20. ] - 3 g2

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

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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 45 5 ' 5 100 9 = 20

0.45 =

3 means 3 ' 8. 8

2. Convert

3 to a decimal. 8

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

Solution 35.5 2 # 100 2 71 = 200

35.5% =

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

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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

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

•

1. Write 0. 4 as a rational number.

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 )

CONTINUED

Try multiplying n by 10. Why doesn’t this work?

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Maths In Focus Mathematics Preliminary Course

Another method 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

This method avoids decimals in the fraction at the end.

(1 ) (2 )

1.3 Exercises 1.

2.

3.

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%

4.

Write each percentage as a decimal. (a) 27% (b) 109% (c) 0.3% (d) 6.23%

5.

Write each fraction as a percentage. 7 20 1 (b) 3 (a)

4 15 1 (d) 1000

(c) 2

6.

Write each decimal as a percentage. (a) 1.24 (b) 0.7 (c) 0.405 (d) 1.2794

7.

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

8.

Write these fractions as recurring decimals. 5 (a) 6 7 (b) 99 13 (c) 99 1 (d) 6 2 (e) 3

Chapter 1 Basic Arithmetic

5 33 1 (g) 7 2 (h) 1 11

31 99 13 + 6 (e) 7+4 (d) 1 -

(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 •

(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

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.

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

2. Evaluate 2

1 ' 3. 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

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4. Evaluate 0.02 # 0.3. Multiply the numbers and count the number of decimal places in the question.

Solution 0.02 # 0.3 = 0.006 5. Evaluate

8.753 . 10

Solution

Move the decimal point 1 place to the left.

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

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.

2.

Write 350 mL as a fraction of 1 litre in its simplest form.

3.

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

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Maths In Focus Mathematics Preliminary Course

4. 5.

6.

Find

3 of $912.60. 5

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

1 of her day 3

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?

8.

John grew by

9.

17 of his height 200 this year. If he was 165 cm tall last year, what is his height now, to the nearest cm?

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

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.

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?

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?

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

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

EXAMPLES 36 = 6 since 6 2 = 36

1.

2.

3

8 = 2 since 2 3 = 8

3.

6

64 = 2 since 2 6 = 64

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=

2

(x 2 - x 1) + (y 2 - y 1)

2

See if you can find other formulae involving indices.

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

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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.

key for square roots.

Use the

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

Use the

3

key for cube roots.

Use the

x

for other roots.

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 =a # # f # a a 14444244443 m + n times = am + n

am ' an = am - n

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

am ' an =

(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

(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 y5

y 18 + (- 4) y5 y

14

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.

3 4

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

Simplify (a) a 6 # a 9 # a 2 (b) y 3 # y - 8 # y 5 (c) a -1 # a -3 1

1

(d) w 2 # w 2 (e) x 6 ' x (f) p 3 ' p - 7 y 11 (g) 5 y

5

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

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

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

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 2

3

(f) x 5 # x 5 m5 # n4 (g) 4 m # n2

Chapter 1 Basic Arithmetic

1

1

p2 # p2

(h)

10. (a) Simplify

2

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

5.

1

2 6 11. Evaluate (a ) when a = c m . 3 12. Evaluate b=

(2m 7) 3

m4 xy 3 #(xy 2) 4 (f) xy 8 4 (2k ) (g) (6k 3) 3 y 12 7 (h) _ 2y 5 i # 8

y=

Evaluate a3b2 when a = 2 and 3 b= . 4

7.

If x = of

8.

2 1 and y = , find the value 3 9

x3 y2 xy 5

.

1 1 1 , b = and c = , 4 2 3 a2 b3 evaluate 4 as a fraction. c

If a =

11

9.

(a) Simplify

8

a b . a8 b7

(b) Hence evaluate a=

a 11 b 8 when a8 b7

5 2 and b = as a fraction. 5 8

x5 y5

when x =

1 and 3

14. Evaluate

k-5 1 when k = . 3 k-9

15. Evaluate

a4 b6 3 when a = and 3 2 2 4 a (b )

b=

6.

x4 y7

2 . 9

-3

p

a3 b6 1 when a = and 2 b4

2 . 3

13. Evaluate

a6 # a4 o a 11 3 5xy 9 x8 # y3

p5 q8 r4

4 3

(d) (7a5b)2

(j) f

.

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

a 8 (b) c m b 4a 3 (c) d 4 n b

(i) e

p4 q6 r2

(b) Hence evaluate

Simplify (a) (pq 3) 5

(e)

p5 q8 r4

1 . 9

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

16. Evaluate

a2 b7 as a fraction in a3 b 2 4 index form when a = c m and 5 5 3 b=c m. 8

17. Evaluate

18. Evaluate

(a 3) 2 b 4 c

as a fraction a (b 2) 4 c 3 6 1 7 when a = , b = and c = . 7 3 9

23

24

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

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

Solution e

0

ab 5 c o . abc 4

0

ab 5 c o =1 abc 4

2. Evaluate 2 - 3 .

Solution 1 23 1 = 8

2-3 =

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

CONTINUED

25

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

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)

1 m3 1 x 1 p7 1 d9 1 k5 1 x2 2 x4 3 y2 1 2z 6 3 5t 8 2 7x

5 2m 6 2 (m) 7 3y (l)

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

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

(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

27

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

1 2

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

2

^ by index laws h

So ` x 2 j = ^ x h = x 1 2

2

1

`

x2 =

x

Now simplify these questions. 1

(a) ^ x 2 h 2 x2

(b)

1 3

(c) ` x 3 j

1

(d) ^ x 3 h 3

3 (e) ^ 3 x h

(f)

3

x3 1 4

4 (g) ` x j

(h) ^ x 4 h 4 1

4 (i) ^ 4 x h

(j)

4

x4

Use your results to complete: 1

xn =

1 n

a =n a

Proof 1 n

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

` a =n a

^ by index laws h

Chapter 1 Basic Arithmetic

EXAMPLES 1. Evaluate (a) 49

1 2 1

(b) 27 3

Solution 1 2

(a) 49 = 49 =7 1 3

(b) 27 = 3 27 =3 2. Write

3x - 2 in index form.

Solution 1

3x - 2 = (3x - 2) 2 1

3. Write (a + b) 7 without fractional indices.

Solution 1

( a + b) 7 = 7 a + b

Putting the fractional and negative indices together gives this rule.

a

1 -n

=

n

1 a

Here are some further rules.

m n

a = n am = (n a ) m

Proof m

1 m

m n

1 n

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

a = ^ am h = n am

29

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 4

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

-

1 3

=

1 1

125 3 1 =3 125 1 = 5

Chapter 1 Basic Arithmetic

-3

(c) c 2 m 3

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

2. Write in index form. x5

(a) (b)

1 (4x - 1) 2 2

3

Solution 5

x5 = x 2 1

(a) (b)

(4x - 1) 2

3

2

=

1 2

(4x 2 - 1) 3 -

= (4x 2 - 1)

3. Write r

-

3 5

2 3

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.

31

32

Maths In Focus Mathematics Preliminary Course

1.7 Exercises 1.

Evaluate (a) 81

3.

1 2

Write without fractional indices. 1

(a) y 3

1

2

(b) 27 3

(b) y 3

1

(c) 16 2

(c) x

1

-

1 2 1

(d) (2x + 5) 2

(d) 8 3 1

(e) (3x - 1)

(e) 49 2 1

-

1 2

1

(f) (6q + r) 3

(f) 1000 3 1

(g) (x + 7)

(g) 16 4

-

2 5

1

(h) 64 2 (i) 64 (j) 1

4.

1 3

(a)

1 7

(k) 81 (l) 32

Write in index form. (b)

t 5

x3

(c)

1 4

(d) (e)

1 5

3

1

(m) 0 8

(f)

(n) 125

1 3

(g)

1 1 1

(r) 9 (s) 8

(i)

(t) 64 2.

(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

3 2 -

1 3 -

2 3

Evaluate correct to 2 decimal places. 1

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

7

5

8

5 .9 # 3 .7 8.79 - 1.4

4

(f)

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

(3x + 1) 5 1

(h)

(q) 256 4

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

(o) 343 3 (p) 128 7

y

5.

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

Chapter 1 Basic Arithmetic

6.

Expand and simplify, and write in index form.

7.

(a) ( x + x) 2 (b) (3 a + 3 b ) (3 a - 3 b ) 1 2 (c) f p + p p 1 2 ) x x ( x 2 - 3x + 1 )

(d) ( x + (e)

x3

33

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

-

-

1 3

2 3

-

4 7

-

2 9

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

-

5 4

6 (3 x + 8 ) (e) 7

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

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

2. Write 7.1#10 -5 as a decimal number.

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

15.4 # 10

-6

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.

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

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

2.

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

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

Remember to put the 0’s in!

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.

7.

Calculate

8.72 #10 - 3 and write 1.34 #10 7 your answer in standard form correct to 3 significant figures.

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

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 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 |

e.g. | 2 # - 3 | = | 2 |#| - 3 | = 6

|a | = a

e.g. | - 3 | 2 = ] - 3 g2 = 9

2

2

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

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 |

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

RHS means Right Hand Side.

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.

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)?

2.

x =1 x 2x = 2x

3.

2x = 2 x

4.

x + y = x+y

5.

2 x = x2

6. 7.

3 x = x3 x +1 = x +1

1.

3x - 2 =1 3x - 2 x 9. =1 x2 10. x $ 0 8.

Discuss absolute value and its definition in relation to these statements.

39

40

Maths In Focus Mathematics Preliminary Course

1.9 Exercises 1.

2.

3.

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

(i) (j)

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.

6.

Show that x 2 = x when (a) x = 5 (b) x = - 2 (c) x = - 3 (d) x = 4 (e) x = - 9.

7.

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

8.

Find values of x for which x = 3.

9.

n Simplify n where n ! 0.

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.

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

x - 2 when x 2 2 x - 2 when x 1 2.

5.

Evaluate a - b if (a) (b) (c) (d) (e)

4.

(h) a + 1 when a 1 -1

x-2 and state which x-2 value x cannot be.

10. Simplify

Chapter 1 Basic Arithmetic

Test Yourself 1 1.

2.

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

6.

(b) (c) (d) (e)

(a) 7 - 2 (b) 5 -1

3.

-

7.

1 2

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

4.5 2 + 7.6 2

(b) 4.3 0.3 2 (c) 3 5.7 (d)

(e) 6 4.

5.

2

(e) 8 3 (f) - 2 - 1

1.3#10 9 3.8 #10 6 -

(g) 49

2 3

-

1 2

as a fraction

1 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

(h) 16 (i) ] -3 g0 (j) 4 - 7 2 - -2 - 3 8.

(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 0

ab 4 o a5 b6

Simplify (a) a 14 ' a 9 6 (b) _ x 5 y 3 i (c) p 6 # p 5 ' p 2

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

Simplify

(e) e

Evaluate (a) - 4 1

Evaluate correct to 3 significant figures. (a)

3 7 5 8 6 2 #3 7 3 3 9' 4 2 1 +2 5 10 5 15# 6

(a) 1

Evaluate as a fraction.

(c) 9

Evaluate

9.

Write in index form. (a)

n 1 (b) 5 x 1 (c) x+y (d)

4

x +1

41

42

Maths In Focus Mathematics Preliminary Course

(e)

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?

a+b

2 (f) x 1 (g) 2x 3 (h)

3

x4

(i)

7

(5x + 3) 9 1

4

m3

(j)

17. Rachel scored 56 out of 80 for a maths test. What percentage did she score?

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

(b) n 4

1

(c) (x + 1) 2 (d) (x - y) -1 (e) (4t - 7) - 4 1

(f) (a + b) 5 (g) x

3

(i) (2x + 3) -

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 x+3 1 (d) (2x - 3) 11

1 3

(h) b 4

(j) x

16. The price of a car increased by 12%. If the car cost $34 500 previously, what is its new price?

4 3

3 2

(c)

6

(e)

3

y7

20. Write in scientific notation. (a) 0.000 013

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%. 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?

15. Johan spends

(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.

Chapter 1 Basic Arithmetic

Challenge Exercise 1 3 2 2 7 + 3 m ' c4 - 1 m. 4 5 3 8

1.

Simplify c 8

2.

3 5 149 7 Simplify + + . 5 12 180 30

3.

4.

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

-

2 3

5.

Write 64

6.

Express 3.2 ' 0.014 in scientific notation correct to 3 significant figures.

7.

as a rational number.

11. Show that 2 (2 k - 1) + 2 k + 1 = 2 (2 k + 1 - 1) . 12. Find the value of

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

25

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.

Vinh scored 17

• • •

a in index form if b3 c2

|x + 1 | x2 - 1

for x ! !1.

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

16. Evaluate 6

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. - 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.

Write 1.3274 as a rational number.

19. Evaluate

9.

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?

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

8.3# 4.1 correct to 0.2 + 5.4 ' 1.3 3 significant figures.

10. Evaluate 3

43