TERMINOLOGY TERMINOLOGY TERMINOLOGY
1
Basic Arithmetic
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. 23 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 ...
Scientifi c 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 7.65 108
# in scientifi c notation
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 signifi cant fi gures, scientifi c 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
Irrational numbers Unreal or imaginary numbers Integers Rational numbers Real numbersIntegers are whole numbers that may be positive, negative or zero. e.g. -4 7 0, , ,-11
Rational numbers can be written in the form of a fraction
b a
where a and b are integers, b!0. e.g. 1 , . , . , 4
3
3 7 0 5• -5
Irrational numbers cannot be written in the form of a fraction
b a (that
is, they are not rational) e.g. 2,r
EXAMPLE
Which of these numbers are rational and which are irrational? 3 1 3, . , , , , . 5 3 9 4 2 65 • r
Solution
3 4
and r are irrational as they cannot be written as fractions (r is irrational).
. , . 1 3 1 3 1 9 1 3 2 65 2 20 13 and •
= = - = - so they are all rational.
Order of operations
1. Brackets: do calculations inside grouping symbols fi rst. (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] +4g.Solution
40 3 (5 4) 40 3 9 40 27 13 # - + = -= -=
PROBLEM
What is wrong with this calculation? Evaluate 1 2 19 4 +- - + Press 19 - 4 ' 1 + 2 = 19 4'1 2 17 What is the correct answer?
BRACKETS KEYS
Use ( and ) to open and close brackets. Always use them in pairs. For example, to evaluate 40-3]5+4g
press 40 3 ( 5 4 ) 3 1 # = - + = To evaluate 1.69 2.77 5.67 3.49 +
correct to 1 decimal place
press : ( ( 5.67 - 3.49 ) ' ( 1.69 + 2.77 ) ) = 0.7 correct to 1 decimal place
=
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
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
-1KEY
Use this key to fi nd the reciprocal of x . For example, to evaluate
7.6 2.1 1 # - 0.063 = -press ( ( ) 7.6- # 2.1 ) x-1 =
(correct to 3 decimal places)
Different calculators use different keys so check the instructions for your calculator.
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.
Solution
.
. .
0 6825371
0 6825371 0 7 correct to 1 decimal place
` # =
2. Round off 0.6825371 correct to 2 decimal places.
Solution
.
. .
0 6825371
0 6825371 0 68 correct to 2 decimal places
` # =
3. Evaluate 3 56. '2 1. correct to 2 decimal places.
Solution
. . . 5
.
3 56 2 1 1 69 238095
1 70 correct to 2 decimal places
' =
= #
Drop off the 2 and all digits to the right as 2 is smaller than 5.
Add 1 to the 6 as the 8 is greater than 5.
Check this on your calculator. Add 1 to the 69 as 5 is too large to just drop off.
While using a fi xed 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 6005232618.60 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
FIX KEY
Use MODE or SET UP to fi x the number of decimal places (see the
instructions for your calculator). This will cause all answers to have a fi xed number of decimal places until the calculator is turned off or switched back to normal.
1. State which numbers are rational and which are irrational.
(a) 169 0.546 (b) (c) -17 (d) 3 r (e) 0 34. • (f) 218 (g) 2 2 (h) 27 1 17.4% (i) (j) 5 1 ch1.indd 7 7/16/09 1:12:20 PM
2. Evaluate (a) 20-8'4 (b) 3#7-2#5 (c) 4#]27' '3g 6 (d) 17+3#-2 (e) 1 9. -2#3 1. (f) 1 3 14'7 - + (g) 2 5 3 5 1 3 2 # - (h) 6 5 1 4 3 8 1 (i) 4 1 8 1 8 5 6 5 ' + (j) 1 4 1 2 1 3 5 1 10 7
3. 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 (f) 4.7 1.3 1 + (g) 4.51 3.28 1 + (h) 5.2 3.6 0.9 1.4 -+ (i) 1.23 3.15 5.33 2.87 -+ (j) 1.72+8.92-3.942
4. Round off 1289 to the nearest hundred.
5. Write 947 to the nearest ten.
6. Round off 3200 to the nearest thousand.
7. A crowd of 10 739 spectators attended a tennis match. Write this fi gure 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.
10. A company made a profi t 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, fi nd the cost of each pie. 18. Evaluate 7.48 correct to 2 decimal places. 19. Evaluate 8 6.4+2.3 correct to 1 decimal place.
20. Find the length of each piece of material, to 1 decimal place, if 25 m of material is cut into 7 equal pieces.
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- =
21. How much will 7.5 m2 of tiles
cost, at $37.59 per m 2 ?
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) 5.6 4.35 3.5 9.8 ++ (d) 7.63 5.12 15.9 6.3 7.8 -+ - (e) 6.87 3.21 1 - 25. Evaluate . . . . 5 39 9 68 5 47 9 91 2 --] g correct to 1 decimal place.
Adding and subtracting
To add: move to the right along the number line To subtract: move to the left along the number line
Subtract Add -4 -3 -2 -1 0 1 2 3 4 Same signs Different signs = + + + = + - = = + = + = -- +
EXAMPLES
Evaluate 1. - +4 3Solution
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
You can also do these on a calculator, or you may have a different way of working these out.
EXAMPLES
Evaluate 1. -2#7Solution
Different signs ( -2and+7 ) give a negative answer.
2# 7 14
- = -
2. -12'-4
Solution
Same signs ( -12and-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 2 = - + = 1 3 - - 1. - +2 3 2. - -7 4 3. 8# -7 4. 7- ]-3g 5. 28' -7 6. -4 9. +3 7. 7. -2 14. -5 37. 8. 4 8. # -7 4. 9. 1 7. - -] 4 87. g 10. 5 3 1 3 2 - - 11. 5-3#4 12. - +2 7#-3 13. 4-3#-2 14. - - -1 2 15. 7+-2 16. 2- -] 1g 17. - +2 15'5 18. -2# #6 -5 19. -28'-7#-5 20. ]-3g2
1.2
Exercises
EvaluateStart at -1 and move 3 places to the right.
Fractions, Decimals and Percentages
EXAMPLES
1. Write 0.45 as a fraction in its simplest form.
Solution
. 0 45 100 45 5 5 20 9 ' = = 2. Convert 8 3 to a decimal.Solution
. . . 8 3 000 0 375 8 3 0 375 So =g
3. Change 35.5% to a fraction.Solution
. % . 35 5 100 35 5 2 2 200 71 # = = 4. Write 0.436 as a percentage.Solution
. . % . % 0 436 0 436 100 43 6 # = =5. Write 20 g as a fraction of 1 kg in its simplest form.
Solution
1kg=1000g 1 20 1000 20 50 1 kg g g g = = Multiply by 100% to change a fraction or decimal to a percentage.Conversions
You can do all theseconversions on your calculator using the
a c b or S+D key. 8 3 means 3'8. ch1.indd 12 7/16/09 1:12:24 PM
Sometimes decimals repeat, or recur. Example . 0. 3 1 0 33333333f 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.
Solution
. ( ) . ( ) ( ) ( ): n n n n 0 44444 1 10 4 44444 2 2 1 9 4 9 4 Let Then f f = = - = = 2. Change 1 329. • • to a fraction.Solution
. ( ) . ( ) ( ) ( ): . . n n n n 1 3292929 1 100 132 9292929 2 2 1 99 131 6 99 131 6 10 10 990 1316 1 495 163 Let Then # f f = = - = = = = A rational number is any number that can be written as a fraction.Check this on your calculator by dividing 4 by 9.
Try multiplying n by 10. Why doesn’t this work? 6. Find the percentage of people who prefer to drink Lemon Fuzzy, if 24
out of every 30 people prefer it.
Solution
% % 30 24 1 100 80 # = CONTINUED ch1.indd 13 7/16/09 1:12:25 PM
1. Write each decimal as a fraction in its lowest terms.
0.64 (a) 0.051 (b) 5.05 (c) 11.8 (d)
2. Change each fraction into a decimal. (a) 5 2 (b) 1 8 7 (c) 12 5 (d) 11 7
3. Convert each percentage to a fraction in its simplest form.
2% (a) 37.5% (b) 0.1% (c) 109.7% (d)
4. Write each percentage as a decimal. 27% (a) 109% (b) 0.3% (c) 6.23% (d)
5. Write each fraction as a percentage. (a) 20 7 (b) 3 1 (c) 2 15 4 (d) 1000 1
6. Write each decimal as a percentage. 1.24 (a) 0.7 (b) 0.405 (c) 1.2794 (d)
7. Write each percentage as a decimal and as a fraction.
52% (a) 7% (b) 16.8% (c) 109% (d) 43.4% (e) (f) 12 % 4 1
8. Write these fractions as recurring decimals. (a) 6 5 (b) 7 99 (c) 99 13 (d) 6 1 (e) 3 2
1.3
Exercises
Another methodLet . . ( ) . ( ) ( ) ( ): n n n n n 1 3292929 10 13 2929292 1 1000 1329 292929 2 2 1 990 1316 990 1316 1 495 163 Then and f f f = = = - = = = This method avoids decimals
in the fraction at the end.
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.
(f) 33 5 (g) 7 1 (h) 1 11 2
9. Express as fractions in lowest terms. (a) 0 8.• (b) 0 2.• (c) 1 5.• (d) 3 7.• (e) 0 67.• • (f) 0 54.• • (g) 0 15. • (h) 0 216. • (i) 0 2 19. • • (j) 1 074. • •
10. Evaluate and express as a decimal. (a) 3 6 5 + (b) 8-3'5 (c) 12 3 4 7 + + (d) 1 99 31 - (e) 7 4 13 6 ++
11. Evaluate and write as a fraction. (a) 7 5. ']4 1. +7 9. g (b) 4.5 1.3 15.7 8.9 -- (c) 12.3 8.9 7.6 6.3 1.7 -+ + (d) . . . 11 5 9 7 4 3 - (e) 8100 64
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.
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 . 5 2 4 3 -Solution
1 5 2 4 3 5 7 4 3 20 28 20 15 20 13 - = -= -= 2. Evaluate 2 2 1 3 ' .Solution
2 2 1 3 2 5 1 3 2 5 3 1 5 6 ' ' # = = = 3. Evaluate 0 056. #100.Solution
. . 0 056#100=5 6 Move the decimal point2 places to the right.
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.
4. Evaluate 0 02. #0 3. .
Solution
. . . 0 02#0 3=0 006 5. Evaluate 10 8.753 .Solution
. . 8 753'10=0 87536. The price of a $75 tennis racquet increased by 5 %. 2 1
Find the new price.
Solution
% $ . $ $ . 5 75 0 055 75 4 13 of ` = # = % . % $ . $ $ . 5 2 1 0 055 105 2 1 75 1 055 75 79 13 2 1 or of # = = =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%)
1% $ . 100% $ . $16.07 112 18 00 112 18 00 1 100 ` # = = =
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 (a) 5 3 4 1 + (b) 3 5 2 2 10 7 - (c) 4 3 1 5 2 # (d) 7 3 4 ' (e) 1 5 3 2 3 2 '
Multiply the numbers and count the number of decimal places in the question.
Move the decimal point 1 place to the left.
4. Find 5 3 of $912.60. 5. Find 7 5 of 1 kg, in grams correct to 1 decimal place. 6. Trinh spends 3 1 of her day sleeping, 24 7 at work and 12 1
eating. What fraction of the day is left?
7. I get $150.00 a week for a casual job. If I spend 10 1 on bus fares, 15 2 on lunches and 3 1 on outings,
how much money is left over for savings?
8. John grew by
200
17 of his height
this year. If he was 165 cm tall last year, what is his height now, to the nearest cm? 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 (i) 100 8.74 (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 profi t of $5 by selling a bike for $85, fi nd the profi t as a percentage of the selling price. 15. Increase 350 g by 15%. 16. Decrease 45 m by 8 %. 2 1
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?
Powers and Roots
A power (or index ) of a number shows how many times a number is multiplied by itself.
PROBLEM
If both the hour hand and minute hand start at the same position at 12 o’clock, when is the fi rst time, correct to a fraction of a minute, that the two hands will be together again?
EXAMPLES
1. 43=4# #4 4=642. 25=2# # # #2 2 2 2=32
In 43 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. 36 =6 since 62=36
2. 3 8=2 since 23=8
3. 6 64 =2 since 26=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 Course book is A=P^1+rhn
Geometry uses formulae involving indices, such as
3 4 V= rr3
. Do you know what this formula is for?
In Chapter 7, the formula for the distance between 2 points on a number plane is
d (x2 x1)2 (y y)
2 1
2
= - + -
See if you can fi nd other formulae involving indices.
Proof
( ) ( ) ( ) a a a a a a a a a a m n a a a m n a 1 times times times m n n m m n ' # # # # # # # # #f f f = = = -=
Index laws
There are some general laws that simplify calculations with indices.
am#an=am n+
Proof
( ) ( ) a a a a a a a a a a a a m n m n m n m n times times times # # # # # # # # # # # f f f = = = ++ 14444 44442 3 14444 44442 3 14444 44442 3
These laws work for any m and n , including fractions and negative numbers. am'an=am n-
(am n) =amn
Proof
( ) ( ) ( ) a a a a a n a n a times times m n m m m m m m m m mn # # # #f = = = f + + + +
POWER AND ROOT KEYS
Use the x2
and x3
keys for squares and cubes. Use the xy or ^ key to fi nd powers of numbers. Use the key for square roots.
Use the 3 key for cube roots.
Use the x for other roots.
(ab)n=a bn n
Proof
( ) ( ) ( ) ( ) ab ab ab ab ab n a a a b b b a b times n n n ntimes ntimes # # # # # # # # # # #f f f = = = 14444 44442 3 14444 44442 3 b a b a n n n = c m
Proof
( ) ( ) ( ) b a b a b a b a b a n b b b b a a a a n n b a times times times n n n # # # # # # # # # # # # f f f = = = c m
EXAMPLES
Simplify 1. m9#m7'm2Solution
m m m m m 9 7 2 9 7 2 14 # ' = = + 2. ( )3 y 2 4Solution
( y ) ( )y y y 2 2 2 8 4 3 3 4 3 3 4 3 12 = = = # CONTINUED ch1.indd 21 7/16/09 1:12:33 PM
1. Evaluate without using a calculator. (a) 53#22 (b) 34+82 (c) 4 1 3 c m (d) 3 27 (e) 4 16
2. Evaluate correct to 1 decimal place. (a) 3.72 (b) 1.061.5 (c) 2.3-0.2 (d) 3 19 (e) 3 34 8. -1 2. #43 1. (f) 0.99 5.61 1 3 + 3. Simplify (a) a6# #a9 a2 (b) y3#y-8#y5 (c) a-1#a-3 (d) w2#w2 1 1 (e) x6'x (f) p3'p-7 (g) y y 5 11 (h) ( )x7 3 (i) (2 )x5 2 (j) (3y-2 4) (k) a3# 'a5 a7 (l) y x 9 2 5 f p (m) w w w 3 6# 7 (n) ( ) p p p 9 2# 3 4 (o) x x x 2 6' 7 (p) ( ) a b a b 4 9 2 2 6 # # (q) ( ) ( ) x y x y 1 4 2 3 3 2 # # 4. Simplify (a) x5#x9 (b) a-1#a-6 (c) m m 3 7 (d) k13# 'k6 k9 (e) a-5# #a4 a-7 (f) x5#x5 2 3 (g) m n m n 4 2 5 4 # #
1.5
Exercises
3. ( ) y y y 5 6 3# -4Solution
( ) y y y y y y y y y y y ( ) 5 6 3 4 5 18 4 5 18 4 5 14 9 # # = = = = - + ch1.indd 22 7/16/09 1:12:34 PM(h) 2 2 p p p 2 # 1 1 (i) (3x11 2) (j) ( ) x x 3 4 6 5. Simplify (a) ( )5 pq3 (b) b a 8 c m (c) 4 b a 4 3 d n (7 (d) a 5 b ) 2 (e) (2 ) m m 4 7 3 (f) xy xy(xy ) 3# 2 4 (g) 3 4 ( ) ( ) k k 6 2 3 8 (h) 2y y 8 5 7 12 # _ i (i) a a a 11 6 4 3 # -e o (j) x y xy 5 8 3 9 3 # f p 6. Evaluate a 3 b 2 when 2 a = and 4 3 b = . 7. If 3 2 x = and 9 1 , y = fi nd the value of xy x y 5 3 2 . 8. If 2 1 , 3 1 a= b= and 4 1 , c = evaluate c a b 4 2 3 as a fraction . 9. (a) Simplify a b a b 8 7 11 8 . Hence evaluate (b) a b a b 8 7 11 8 when 5 2 a = and 8 5 b = as a fraction . 10. (a) Simplify p q r p q r 4 6 2 5 8 4 . (b) Hence evaluate p q r p q r 4 6 2 5 8 4 as a fraction when 8 7, 3 2 p= q= and 4 3 r = . 11. Evaluate ( )a4 3 when 6 . a 3 2 = 1 c m 12. Evaluate b a b 4 3 6 when a 2 1 = and b 3 2 = . 13. Evaluate x y x y 5 5 4 7 when x 3 1 = and y 9 2 = . 14. Evaluate k k 9 5 when k . 3 1 = 15. Evaluate ( ) a b a b 3 2 2 4 6 when a=43 and b 9 1 = . 16. Evaluate a b a b 5 2 6 3 # # as a fraction when a 9 1 = and b 4 3 = . 17. Evaluate a b a b 3 2 7 as a fraction in index form when a
5 2 4 = c m and b 8 5 3 = c m . 18. Evaluate ( ) ( ) a b c a b c 2 4 3 3 2 4 as a fraction when a , 3 1 = b 7 6 = and c 9 7 = .
ch1.indd 23 7/16/09 1:12:35 PM
Proof
x x x x x x x x x 1 1 n n n n n n n n 0 0 ' ' ` = = = = =
Negative and zero indices
Class Investigation
Explore zero and negative indices by looking at these questions. For example simplify x3'x5 using (i) index laws and (ii) cancelling.
(i) x3'x5=x-2 by index laws
(ii) x x x x x x x x x x x 1 5 3 2 # # # ## # = = x x 1 So 2 2 = -
Now simplify these questions by (i) index laws and (ii) cancelling. (a) x2'x3 (b) x2'x4 (c) x2'x5 (d) x3'x6 (e) x3'x3 (f) x2'x2 (g) x'x2 (h) x5'x6 (i) x4'x7 (j) x'x3
Use your results to complete: x x 0 n = = - x0=1
ch1.indd 24 7/16/09 1:12:36 PM
1 x x n n = -
Proof
x x x x x x x x x x x 1 1 n n n n n n n n 0 0 0 0 ' ' ` = = = = =
EXAMPLES
1. Simplify . abc ab c 4 5 0 e oSolution
1 abc ab c 4 5 0 = e o 2. Evaluate 2-3.Solution
2 2 1 8 1 3 3 = =3. Write in index form. (a) 1 x2 (b) 3 x5 (c) 5 1 x (d) x 1 1 +
CONTINUED ch1.indd 25 7/16/09 1:12:37 PM
1. Evaluate as a fraction or whole number. (a) 3-3 (b) 4-1 (c) 7-3 (d) 10-4 (e) 2-8 6 (f) 0 (g) 2-5 (h) 3-4 (i) 7-1 (j) 9-2 (k) 2-6 (l) 3-2 4 (m) 0 (n) 6-2 (o) 5-3 (p) 10-5 (q) 2-7 (r) 20 (s) 8-2 (t) 4-3 2. Evaluate (a) 20 (b) 2 1 -4 c m (c) 3 2 -1 c m (d) 6 5 -2 c m (e) 3 2 x y x y 0 -+ f p (f) 5 1 -3 c m (g) 4 3 -1 c m (h) 7 1 -2 c m (i) 3 2 -3 c m (j) 2 1 -5 c m (k) 7 3 -1 c m
1.6
Exercises
Solution
(a) 1 x2 x 2 = - (b) x x x 3 3 1 3 5 5 5 # = = (c) x x x 5 1 5 1 1 5 1 1 # = = (d) ( ) x x x 1 1 1 1 1 1 1 + = + =] + g4. Write a −3 without the negative index.
Solution
a a 1 3 3 = -ch1.indd 26 7/16/09 1:12:38 PM
(l) 9 8 0 c m (m) 7 6 -2 c m (n) 10 9 -2 c m (o) 11 6 0 c m (p) 4 1 2 - -c m (q) 5 2 3 - -c m (r) 3 7 2 1 - -c m (s) 8 3 0 -c m (t) 1 4 1 2 - -c m
3. Change into index form. (a) 1 m3 (b) 1x (c) 1 p7 (d) 1 d9 (e) 1 k5 (f) 1 x2 (g) 2 x4 (h) 3 y2 (i) 2 1 z6 (j) 5 3 t8 (k) 7 2 x (l) 2 5 m6 (m) 3 2 y7 (n) (3 4) 1 x+ 2 (o) ( ) 1 a+b 8 (p) 2 1 x - (q) (5p 1) 1 3 + (r) (4 9) 2 t- 5 (s) (x ) 4 1 1 11 + (t) 9 ( 3 ) 5 a+ b 7
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) ] g2x-3 (j) ]4ng-1 (k) ]x+1g-6 (l) ^8y+zh-1 (m) ]k-3g-2 (n) ^3x+2yh-9 (o) 1x 5 -b l (p) 1y 10 -c m (q) 2 p 1 -d n (r) 1 a b 2 + -c m (s) xx yy 1 -+ -e o (t) 3 2 x y w z 7 + -e o
ch1.indd 27 7/16/09 1:12:39 PM
Proof
n n a a a a a a by index laws n n n n ` = = = 1 1 ` ^ ^ j h h
Fractional indices
Class Investigation
Explore fractional indices by looking at these questions. For example simplify (i) 2
x 2 1 ` j and (ii) ^ xh2. 2 ( ) x x x i by index laws 2 1 = = 1 ` j ^ h 2 2 ( ) x x x x x x x ii So 2 2 2 ` = = = = 1 1 ^ ` ^ h j h
Now simplify these questions. (a) x2 2 1 ^ h (b) x2 (c) 3 x 3 1 ` j (d) x3 3 1 ^ h (e) ^3 xh3 (f) 3 x3 (g) x4 4 1 ` j (h) x4 4 1 ^ h (i) ^4 xh4 (j) 4 x4
Use your results to complete: n x = 1 n a1=n a ch1.indd 28 7/16/09 1:12:40 PM
EXAMPLES
1. Evaluate (a) 492 1 (b) 273 1Solution
(a) 492 49 7 = = 1 (b) 273 27 3 3 = = 12. Write 3x-2 in index form.
Solution
( )2 x x 3 - =2 3 -2 1 3. Write (a+b)7 1without fractional indices.
Solution
7 (a+b)1 =7 a+bProof
n n n n a a a a a m n m m m n = = a = = m m 1 1 ` ^ ^ j h h
Putting the fractional and negative indices together gives this rule.
- n a a 1 n = 1
Here are some further rules.
n ( ) a a a m n n m = = m
ch1.indd 29 7/16/09 1:12:41 PM
b a a b n n = -c m b l
EXAMPLES
1. Evaluate (a) 834 (b) 125-3 1 (c) 3 2 -3 c mSolution
(a) 83 ( 8) ( 8 ) 2 16 or 3 4 3 4 4 = = = 4 (b) -3 3 125 125 1 125 1 5 1 3 = = = 1 1Proof
b a b a b a b a a b a b a b 1 1 1 1 n n n n n n n n n n n ' # = = = = = = -c c b m m l ch1.indd 30 7/16/09 1:12:42 PM
(c) 3 2 2 3 8 27 3 8 3 3 3 = = = -c m c m
2. Write in index form. (a) x5 (b) (4x 1) 1 2 2 3 -
Solution
(a) 2 x5 =x 5 (b) -3 3 ( ) ( ) ( ) x x x 4 1 1 4 1 1 4 1 2 2 3 2 2 - = -= -2 23. Write -5 r 3
without the negative and fractional indices.
Solution
-5 5 r r r 1 1 3 5 = = 3 3DID YOU KNOW?
Nicole Oresme (1323–82) was the fi rst mathematician to use fractional indices.
John Wallis (1616–1703) was the fi rst person to explain the signifi cance of zero, negative
and fractional indices. He also introduced the symbol 3 for infi nity.
Do an Internet search on these mathematicians and fi nd out more about their work and backgrounds. You could use keywords such as indices and infi nity as well as their names to fi nd this information.
1. Evaluate (a) 8121 (b) 273 1 (c) 1621 (d) 83 1 (e) 4921 (f) 10003 1 (g) 1641 (h) 642 1 (i) 6431 (j) 17 1 (k) 8141 (l) 325 1 (m) 081 (n) 1253 1 (o) 34331 (p) 1287 1 (q) 25641 (r) 92 3 (s) 8-31 (t) 64-3 2
2. Evaluate correct to 2 decimal places. (a) 234 1 (b) 4 45.8 (c) 7 1.24+4.32 (d) 12.9 1 5 (e) . . . . 1 5 3 7 3 6 1 4 8 + - (f) . . . . 8 79 1 4 5 9 3 7 4 # -
3. Write without fractional indices. (a) 3 y 1 (b) y3 2 (c) 2 x- 1 (d) (2x+5)2 1 (e) ( )-2 x 3 -1 1 (f) (6q+r)3 1 (g) ( )-5 x+7 2
4. Write in index form. (a) t (b) 5 y (c) x3 (d) 3 9-x (e) 4s+1 (f) 2 3 1 t + (g) (5 ) 1 x-y 3 (h) (3x+1)5 (i) ( 2) 1 x 2 3 - (j) 2 7 1 y + (k) 4 5 x 3 + (l) y 3 1 2 2- (m) 5 ( 2) 3 x2 3 4 +
5. Write in index form and simplify. (a) x x (b) xx (c) x x 3 (d) x x 3 2 (e) x x4
1.7
Exercises
ch1.indd 32 7/16/09 1:12:43 PM
6. Expand and simplify, and write in index form. (a) ( x+x)2 (b) (3 a+3 b) (3 a-3 b) (c) p 1 p 2 + f p (d) ( x 1 ) x 2 + (e) ( ) x x x 3x 1 3 2- +
7. Write without fractional or negative indices. (a) (a-2b)-3 1 (b) (y-3)- 23 (c) 4 6( a+1)-7 4 (d) -4 (x y) 3 + 5 (e) -9 ( x ) 7 6 3 +8 2
Scientifi c notation (standard form)
Very large or very small numbers are usually written in scientifi c notation to make them easier to read. What could be done to make the fi gures 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 scientifi c notation is written as a number between 1 and 10 multiplied by a power of 10.
EXAMPLES
1. Write 320 000 000 in scientifi c notation.
Solution
.
320 000 000=3 2#108
2. Write 7 1. #10-5 as a decimal number.
Solution
. . . 7 1 10 7 1 10 0 000 071 5 5 # = ' =Write the number between 1 and 10 and count the decimal places moved.
Count 5 places to the left.
SIGNIFICANT FIGURES
The concept of signifi cant fi gures is related to rounding off. When we look at very large (or very small) numbers, some of the smaller digits are not signifi cant.
For example, in a football crowd of 49 976, the 6 people are not really signifi cant in terms of a crowd of about 50 000! Even the 76 people are not signifi cant.
When a company makes a profi t of $5 012 342.87, the amount of 87 cents is not exactly a signifi cant sum! Nor is the sum of $342.87.
To round off to a certain number of signifi cant fi gures, we count from the fi rst non-zero digit.
In any number, non-zero digits are always signifi cant. Zeros are not signifi cant, except between two non-zero digits or at the end of a decimal number.
Even though zeros may not be signifi cant, they are still necessary. For example 31, 310, 3100, 31 000 and 310 000 all have 2 signifi cant fi gures but are very different numbers!
Scientifi c notation uses the signifi cant fi gures in a number.
SCIENTIFIC NOTATION KEY
Use the EXP or #10x key to put numbers in scientifi c notation. For example, to evaluate 3.1 10# 4'2.5#10 ,-2
press 3.1 EXP 4 2.5 EXP ( ) 2 1240 000
' - =
=
DID YOU KNOW?
Engineering notation is similar to scientifi c notation, except the powers of 10 are always multiples of 3. For example,
3.5#103 15.4#10-6
EXAMPLES
. ( ) . . ( ) . . ( ) 12 000 1 2 10 2 0 000 043 5 4 35 10 3 0 020 7 2 07 10 3 significant figures significant figures significant figures 4 5 2 # # # = = =When rounding off to signifi cant fi gures, use the usual rules for rounding off.
EXAMPLES
1. Round off 4 592 170 to 3 signifi cant fi gures.
Solution
4 592 170=4 590 000 to 3 signifi cant fi gures
2. Round off 0.248 391 to 2 signifi cant fi gures.
Solution
. .
0 248 391=0 25 to 2 signifi cant fi gures
3. Round off 1.396 794 to 3 signifi cant fi gures.
Solution
. .
1 396 794=1 40 to 3 signifi cant fi gures
1. Write in scientifi c notation . 3 800 (a) 1 230 000 (b) 61 900 (c) 12 000 000 (d) 8 670 000 000 (e) 416 000 (f) 900 (g) 13 760 (h) 20 000 000 (i) 80 000 (j)
2. Write in scientifi c notation. 0.057 (a) 0.000 055 (b) 0.004 (c) 0.000 62 (d) 0.000 002 (e) 0.000 000 08 (f) 0.000 007 6 (g) 0.23 (h) 0.008 5 (i) 0.000 000 000 07 (j)
3. Write as a decimal number. (a) 3 6. #104 (b) 2 78. #107 (c) 9 25. #103 (d) 6 33. #106 (e) 4#105 (f) 7 23. #10-2 (g) 9 7. #10-5 (h) 3 8. #10-8 (i) 7#10-6 (j) 5#10-4
4. Round these numbers to 2 signifi cant fi gures.
235 980 (a) 9 234 605 (b) 10 742 (c) 0.364 258 (d) 1.293 542 (e) 8.973 498 011 (f) 15.694 (g) 322.78 (h) 2904.686 (i) 9.0741 (j)
1.8
Exercises
Remember to put the 0’s in! ch1.indd 35 7/16/09 1:12:46 PM
5. Evaluate correct to 3 signifi cant fi gures. (a) 14 6. #0 453. (b) 4 8. '7 (c) 4.47+2 59. #1 46. (d) . . 3 47 2 7 1 - 6. Evaluate 4 5. #104#2 9. #105,
giving your answer in scientifi c notation. 7. Calculate . . 1 34 10 8 72 10 7 3 # # and write your answer in standard form correct to 3 signifi cant fi gures.
Investigation
A logarithm is an index. It is a way of fi nding the power (or index) to which a base number is raised. For example, when solving 3x=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.
If ax=y then log y x a =
The expression log
1. 7 49 means the power of 7 that gives 49.
The solution is 2 since 72=49.
The expression log
2. 2 16 means the power of 2 that gives 16.
The solution is 4 since 24=16.
Can you evaluate these logarithms?
log 1. 3 27 log 2. 5 25 log 3. 10 10 000 log 4. 2 64 log 5. 4 4 log 6. 7 7 log 7. 3 1 log 8. 4 2 9. 3 1 log3 10. 4 1 log2 The a is called the base
number and the x is the
index or power.
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 c2=9 gives two solutions,
c=!3 .
However when solving c2=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 defi nitions
We write the absolute value of x as xx x x x 0 when when x 01 $ = -)
EXAMPLES
1. Evaluate 4 .Solution
4 4 = since4$0We can also defi ne
x as the distance of x from 0 on the number line. We will use this in Chapter 3.
CONTINUED
2. Evaluate -3 .
Solution
3 3 3 0 3 since 1 - = - - -= ] gThe absolute value has some properties shown below.
Properties of absolute value
a 9 = = = | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ab a b a a a a a a b b a a b a b 2 3 2 3 6 3 3 5 5 5 7 7 7 2 3 3 2 1 2 3 2 3 3 4 3 4 e.g. e.g. e.g. e.g. e.g. e.g. but 2 2 2 2 2 2 # # # 1 # = - = - = - -= = = - = - = = - = - - = - = + + + = + - + - + ] g
EXAMPLES
1. Evaluate 2 - - + -1 32.Solution
2 1 3 2 1 3 2 1 9 10 2 2 - - + - = - + = - + =2. Show that a+b # a + b when a= -2 and b =3.
Solution
a b 2 3 1 1 LHS= + = - + = = LHS means Left Hand Side.a b 2 3 2 3 5 RHS= + = - + = + = a b a b 1 5 Since 1 # + +
3. Write expressions for 2x -4 without the absolute value signs.
Solution
1 x x x x x x x x x x x 2 4 2 4 2 4 0 2 4 2 2 4 2 4 2 4 0 2 4 2 4 2 when i.e. when i.e. 1 1 $ $ $ - = - -- = - - -= - + ] gClass Discussion
Are these statements true? If so, are there some values for which the expression is undefi ned (values of x or y that the expression cannot have)? 1. x x =1 2. 2x =2x 3. 2x =2 x 4. x + y = x+y 5. x2=x2 6. x3=x3 7. x+1 = x +1 8. x x 3 2 3 2 1 -- = 9. x x 1 2 = 10. x $0
Discuss absolute value and its defi nition in relation to these statements.
RHS means Right Hand Side.
1. Evaluate (a) 7 (b) -5 (c) -6 (d) 0 (e) 2 (f) -11 (g) -2 3 (h) 3 -8 (i) -52 (j) -53 2. Evaluate (a) 3 + -2 (b) - -3 4 (c) - +5 3 (d) 2#-7 (e) -3 + -1 (f) 5- -2 # 6 2 (g) - +2 5#-1 (h) 3 -4 (i) 2 - -3 3 -4 (j) 5-7 +4 -2 3. Evaluate a-b if (a) a=5 2andb= (b) a= -1andb=2 (c) a= -2andb= -3 (d) a=4 7andb= (e) a= -1andb= -2.
4. Write an expression for (a) a whena20 (b) a whena10 (c) a whena=0 (d) 3a whena20 (e) 3a whena10 (f) 3a whena=0 (g) a+1 whena2-1 (h) a+1 whena1-1 (i) x-2 whenx22 (j) x-2 whenx12 . 5. Show that a+b # a + b when (a) a=2 4andb= (b) a= -1andb= -2 (c) a= -2 3andb= (d) a= -4 5andb= (e) a= -7andb= -3.
6. Show that x2 = x when
(a) x =5 (b) x= -2 (c) x= -3 (d) x =4 (e) x= -9.
7. Use the defi nition 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. Simplify nn where n!0. 10. Simplify 2 2 x x
-- and state which
value x cannot be.
1.9
Exercises
1. Convert 0.45 to a fraction (a) 14% to a decimal (b) (c) 8 5 to a decimal 78.5% to a fraction (d) 0.012 to a percentage (e) (f) 15 11 to a percentage 2. Evaluate as a fraction. (a) 7-2 (b) 5-1 (c) 9- 12
3. Evaluate correct to 3 signifi cant fi gures.
(a) 4 5. 2+7 6. 2 (b) 4.30.3 (c) 5.7 2 3 (d) . . 3 8 10 1 3 10 6 9 # # (e) 6-3 2 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 5. Simplify (a) x5# 'x7 x3 (b) (5 )y3 2 (c) ( ) a b a b 9 5 4 7 (d) 3 2x6 3 d n (e) a b ab 5 6 4 0 e o 6. Evaluate (a) 1 5 3 8 7 - (b) 7 6 3 3 2 # (c) 9 4 3 ' (d) 5 2 2 10 1 + (e) 15 6 5 # 7. Evaluate (a) -4 (b) 362 1 (c) -5 2-2 3 (d) 4-3 as fraction (e) 83 2 (f) - -2 1 (g) 49- 12 as a fraction (h) 164 1 (i) ]-3g0 (j) 4-72- - -2 3 8. Simplify (a) a14'a9 (b) _x y5 3 6i (c) p6# 'p5 p2 (d) ^2b9 4h (e) (2 ) x y x y 10 7 3 2
9. Write in index form.
(a) n (b) 1 x5 (c) 1 x+y (d) 4 x+1
Test Yourself 1
ch1.indd 41 7/16/09 1:12:52 PM(e) 7 a+b (f) 2x (g) 2 1 x3 (h) 3 x4 (i) 7 (5x+3)9 (j) 1 m3 4
10. Write without fractional or negative
indices . (a) a-5 (b) n4 1 (c) (x+1)2 1 (d) (x-y)-1 (e) (4t-7)-4 (f) (a+b)5 1 (g) x- 13 (h) b4 3 (i) (2x+3)3 4 (j) x-2 3
11. Show that a+b # a + b when a =5
and b = -3 . 12. Evaluate a 2 b 4 when 25 9 a = and 1 3 2 b = . 13. If 3 1 a 4 = c m and 4 3, b = evaluate ab3 as a fraction. 14. Increase 650 mL by 6%. 15. Johan spends 3
1 of his 24-hour day
sleeping and
4
1 at work.
How many hours does Johan spend (a)
at work?
What fraction of his day is spent at (b)
work or sleeping?
If he spends 3 hours watching TV, (c)
what fraction of the day is this?
What percentage of the day does he (d)
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
scientifi c notation correct to 1 decimal place.
19. Write in index form.
(a) x (b) 1y (c) 6 x+3 (d) (2 3) 1 x- 11 (e) 3 y7
20. Write in scientifi c notation.
0.000 013 (a) 123 000 000 000 (b)
21. Convert to a fraction. (a) 0 7.• (b) 0 124. • •
22. Write without the negative index.
(a) x-3 (b) (2a+5)-1 (c) b a -5 c m
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
2
a = - and b = -5.
1. Simplify 8 4 3 3 3 2 4 1 . 5 2 8 7 ' + -c m c m 2. Simplify . 5 3 12 5 180 149 30 7 + + -
3. Arrange in increasing order of size: 51%, 0.502, 0 5. ,• . 99 51 4. Mark spends 3
1 of his day sleeping, 12
1
of the day eating and
20 1
of the day watching TV. What percentage of the day is left?
5. Write 64-32 as a rational number.
6. Express 3 2. 25'0 014. in scientifi c
notation correct to 3 signifi cant fi gures.
7. Vinh scored 17
2
1 out of 20 for a maths
test, 19 out of 23 for English and 55 2 1
out of 70 for physics. Find his average score as a percentage, to the nearest whole percentage.
8. Write 1 3274. • • • as a rational number.
9. The distance from the Earth to the moon is 3 84. #105 km. How long would it take
a rocket travelling at 2 13. #10 km h4 to
reach the moon, to the nearest hour? 10. Evaluate . . . . . 0 2 5 4 1 3 8 3 4 1 3 ' # + correct to
3 signifi cant fi gures.
11. Show that 2 2( k-1)+2k+1=2 2( k+1-1) .
12. Find the value of
b c a 3 2 in index form if . , a b c 5 2 3 1 5 3 and 4 3 2 =c m = -c m =c m
13. Which of the following are rational numbers: 3, 0 34 2 3. , , , . , ,1 5 0 7 3 • r - ?
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 x 1 1 2 -+ for x!!1. 16. Evaluate 2.4 3.31 4.3 2.9 3 2 1.3 6 + - correct to 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 signifi cant fi gures. 19. Evaluate ( . ) . . 6 9 10 3 4 10 1 7 10 5 3 3 2 # # # - - + and express your answer in scientifi c notation correct to 3 signifi cant fi gures.
20. Prove |a+b|#|a|+|b| for all real a , b .
Challenge Exercise 1
TERMINOLOGY
2
Algebra and
Surds
Binomial: A mathematical expression consisting of
two terms such as x + or x3 3 -1
Binomial product: The product of two binomial
expressions such as (x+3) (2x-4)
Expression: A mathematical statement involving numbers,
pronumerals and symbols e.g. x2 - 3
Factorise: The process of writing an expression as a
product of its factors. It is the reverse operation of expanding brackets i.e. take out the highest common factor in an expression and place the rest in brackets e.g. 2y -8=2 (y-4)
Pronumeral: A letter or symbol that stands for a number
Rationalising the denominator: A process for replacing a
surd in the denominator by a rational number without altering its value
Surd: From ‘absurd’. The root of a number that has an
irrational value e.g. 3 . It cannot be expressed as a rational number
Term: An element of an expression containing
pronumerals and/or numbers separated by an operation such as ,+ -,#or' e.g. 2 ,x -3
Trinomial: An expression with three terms such as
x x
3 2 2 1
- +
DID YOU KNOW?
Box text...
INTRODUCTION
THIS CHAPTER REVIEWS ALGEBRA skills, including simplifying expressions, removing grouping symbols, factorising, completing the square and
simplifying algebraic fractions . Operations with surds , including rationalising the denominator , are also studied in this chapter .
DID YOU KNOW?
One of the earliest mathematicians to use algebra was Diophantus of Alexandria . It is not known
when he lived, but it is thought this may have been around 250 AD.
In Baghdad around 700–800 AD a mathematician named Mohammed Un-Musa Al-Khowarezmi wrote books on algebra and Hindu numerals. One of his books was named
Al-Jabr wa’l Migabaloh , and the word algebra comes from the fi rst word in this title.
