Mq 10 Surds and Indices

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18A Number classification review18B Surds18C Operations with surds18D Fractional indices18E Negative indices

18F Logarithms18G Logarithm laws18H Solving equations

WHAT DO YOU KNOW?

1 List what you know about real numbers.Create a concept map to show your list.

2 Share what you know with a partner andthen with a small group.

3 As a class, create a large concept mapthat shows your classs knowledge ofreal numbers.

18

OPENING QUESTION

Mannings formula is a formula used to

estimate the flow of water down a river in

a flood event, measured in metres per

second. The formula is v R S

n==

2

3

1

2

, where

Ris the hydraulic radius, Sis the slope

of the river and nis the roughness

coefficient. What will be the flow of

water in the river ifR

= 8,S

= 0.0025 andn= 0.625?

NUMBER AND ALGEBRA REAL NUMBERS

Real numbers


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590 Maths Quest 10 +10A for the Australian Curriculum

Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be obtained by

completing the matching SkillSHEET. Either search for the SkillSHEET in your eBookPLUS or

ask your teacher for a copy.

Identifying surds

1 Which of the following are surds?a 7 b 10

c 49 d 4 2

Simplifying surds

2 Simplify each of the following.

a 48 b 98

c 5 12 d 3 72

Adding and subtracting surds


a 2 6 4 3 7 3 5 6 + b 2 32 5 45 4 180 10 8 +

Multiplying and dividing surds


a 7 10 b 2 3 4 6

c6

2

d5 6

10 3

Evaluating numbers in index form

5 Evaluate each of the following.

a 72 b 34

c (2.5)6 d (0.3)4

Using the index laws


a x3x7 b 4y35y8

c 24a3b6ab5 d (2m4)2

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5Chapter 18 Real numbers

Number classification review The number systems used today evolved from a basic and practical need of primitive people

to count and measure magnitudes and quantities such as livestock, people, possessions, time

and so on. As societies grew and architecture and engineering developed, number systems became more

sophisticated. Number use developed from solely whole numbers to fractions, decimals and

irrational numbers.

The Real Number System contains the set of rational and irrational numbers. It is denoted by

the symbolR. The set of real numbers contains a number of subsets which can be classified asshown in the chart below.

Real numbers R

Irrational numbers I

(surds, non-terminating

and non-recurring

decimals, ,e)

Rational numbers Q

Integers

Z

Non-integer rationals

(terminating and

recurring decimals)

Zero

(neither positive

nor negative)

Positive

(Natural

numbers N)

Z+Negative

Z

Rational numbers (Q) A rational number(ratio-nal) is a number that can be expressed as a ratio of two whole

numbers in the forma

b, where b0.

Rational numbers are given the symbol Q. Examples are:

1

5, 27, 310

, 94, 7, -6, 0.35, 1 4.

i

18A

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Interactivity

Classifying

numbers

int-2792


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Integers (Z) Rational numbers may be expressed as integers. Examples are:

5

1=5,

4

1=-4,

27

1=27, -

15

1=-15

The set of integers consists of positive and negative whole numbers and 0 (which is neither

positive nor negative). They are denoted by the letterZand can be further divided into

subsets. That is: Z={. . ., -3, -2, -1, 0, 1, 2, 3, . . .}

Z +={1, 2, 3, 4, 5, 6, . . .}

Z -={-1, -2, -3, -4, -5, -6, . . .}

Positive integers are also known as natural numbers(or counting numbers) and are denoted

by the letterN. That is:

N={1, 2, 3, 4, 5, 6, . . .}

Integers may be represented on the number line as illustrated below.

-3 -2 -1 3 Z210

The set of integers

N

The set of positive integersor natural numbers

1 2 3 4 5 6 Z-

The set of negative integers-1-2-3- 4-5-6

Note:Integers on the number line are marked with a solid dot to indicate that they are the

only points in which we are interested.

Non-integer rationals Rational numbers may be expressed as terminating decimals. Examples are:

7

10=0.7,

1

4=0.25,

5

8=0.625,

9

5=1.8

These decimal numbers terminate after a specific number of digits. Rational numbers may be expressed as recurring decimals(non-terminating or periodic

decimals). For example:

1

3=0.333 333 . . . or 0 3.

9

11=0.818 181 . . . or 0 81. (or 0 81. )

5

6=0.833 333 . . . or 0 83.

3

13=0.230 769 230 769 . . . or 0 230769. (or 0 230769. )

These decimals do not terminate, and the specific

digit (or number of digits) is repeated in a pattern.

Recurring decimals are represented by placing a dot or

line above the repeating digit or pattern.

Rational numbers are defined in set notation as:Q=rational numbers

Q a

ba b Z b=

, , , 0 where means an element of.

Irrational numbers (I) An irrational number(ir-ratio-nal) is a number that cannot be expressed as a ratio of two

whole numbers in the forma

b, where b0.

Irrational numbers are given the symbolI. Examples are:

7, 13, 5 21,7

9, p, e

-3-4 -2 -1 3210 4

-3.743

1.63 3.6-234

12

Q


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Irrational numbers may be expressed as decimals. For example:

5=2.236 067 977 5 . . . 0 03. =0.173 205 080 757 . . .

18=4.242 640 687 12 . . . 2 7=5.291 502 622 13 . . .

p =3.141 592 653 59 . . . e=2.718 281 828 46 . . .

These decimals do not terminate, and the digits do not repeat themselves in any particular

pattern or order (that is, they are non-terminating and non-recurring). Rational and irrational numbers belong to the set ofreal numbers (denoted by the symbolR). They can

be positive, negative or 0. The real numbers may be

represented on a number line as shown at right (irrational

numbers above the line; rational numbers below it). To classify a number as either rational or irrational:

1. Determine whether it can be expressed as a whole number, a fraction or a terminating or

recurring decimal.

2. If the answer is yes, the number is rational; if the answer is no, the number is irrational.

p(pi) The symbol p(pi) is used for a particular number; that is, the circumference of a circle whose

diameter length is 1 unit. It can be approximated as a decimal that is non-terminating and non-recurring. Therefore, p

is classified as an irrational number. (It is also called a transcendental number and cannot be

expressed as a surd.) In decimal form, p =3.141 592 653 589 793 23 . . . It has been calculated to

29 000 000 (29 million) decimal places with the aid of a computer.

Specify whether the following numbers are rational or irrational.

a1

5 b 25 c 13 d 3p e 0.54 f 643 g 323 h 1

27

3

THINK WRITE

a 1

5is already a rational number. a

1

5is rational.

b 1 Evaluate 25. b 25 5=

2 The answer is an integer, so classify 25. 25is rational.

c 1 Evaluate 13. c 13=3.605 551 275 46 . . .

2 The answer is a non-terminating andnon-recurring decimal; classify 13.

13is irrational.

d 1 Use your calculator to find the value of

3p.

d 3p =9.424 777 960 77 . . .

2 The answer is a non-terminating and

non-recurring decimal; classify 3p.

3p is irrational.

e 0.54 is a terminating decimal; classify it

accordingly.

e 0.54 is rational.

f 1 Evaluate 643 . f 643 =4

2The answer is a whole number, soclassify 643 .

643 is rational.

-3-4 -2 -1 3210 4

12

- 4

- 12 - 5 2

R

WORKED EXAMPLE 1


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g 1 Evaluate 323 . g 323 =3.174 802 103 94 . . .

2 The result is a non-terminating and

non-recurring decimal; classify 323 .

323 is irrational.

h 1 Evaluate1

27

3 . h1

27

1

3

3=

2 The result is a number in a rational form.1

27

3 is rational.

REMEMBER

1. Rational numbers (Q) can be expressed in the forma

b, where aand bare whole

numbers and b0. They include whole numbers, fractions and terminating and

recurring decimals.

2. Irrational numbers (I) cannot be expressed in the form

a


numbers and b0. They include surds, non-terminating and non-recurring decimals,

and numbers such as pand e.

3. Rational and irrational numbers together constitute the set of real numbers (R).

Number classification reviewFLUENCY

1 WE 1 Specify whether the following numbers are rational (Q) or irrational (I ).

a 4 b5

c7

9d 2 e 7

f 0 04. g 21

2h 5 i

9

4j 0.15

k -2.4 l 100 m 14 4. n 1 44. o p

p25

9q 7.32 r 21 s 1000 t 7.216 349 157 . . .

u 81 v 3p w 623 x1

16y 0 00013 .

2 Specify whether the following numbers are rational (Q), irrational (I ) or neither.a

1

8b 625 c

11

4d

0

8e -6

1

7

f 813 g 11 h1 44

4

.i j

8

0

k 213 l

7m ( )5

23n -

3

11o

1

100

p64

16q

2

25r

6

2s 273 t

1

4

u22

7

v

1 7283

. w 6 4 x 4 6 y 2

4

( )

EXERCISE

18A


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3 MC Which of the following best represents a rational number?

A p B4

9C

9

12D 33 E 5

4 MC Which of the following best represents an irrational number?

A 81 B6

5C 3433 D 22 E 12

5 MC Which of the following statements regarding the numbers -0.69, 7,

3, 49is correct?

A

3is the only rational number.

B 7and 49are both irrational numbers.

C -0.69 and 49are the only rational numbers.

D -0.69 is the only rational number.

E 7is the only rational number.

6 MC Which of the following statements regarding the numbers 21

2, -

11

3, 624, 993 is correct?

A-11

3 and624

are both irrational numbers.

B 624 is an irrational number and 993 is a

rational number.

C 624 and 993 are both irrational numbers.

D 21

2is a rational number and -

11

3is an irrational

number.

E 993 is the only rational number.

Surds A surdis an irrational number that is represented by a root sign or a radical sign, forexample: , 3 , 4

Examples of surds include: 7, 5, 113 , 154

Examples that are not surds include:

9, 16 , 1253 , 814

Numbers that are not surds can be simplified to rational numbers, that is:

9 3= , 16 4= , 125 53 = , 81 34 =

Which of the following numbers are surds?

a 16 b 13 c1

16 d 173 e 634 e 17283

THINK WRITE

a 1 Evaluate 16 . a 16 4=

2The answer is rational (since it is a wholenumber), so state your conclusion.

16is not a surd.

REFLECTIONWhy is it important to understand

the real number system?

18B

WORKED EXAMPLE 2


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b 1 Evaluate 13. b 13=3.605 551 275 46 . . .

2 The answer is irrational (since it is a non-

recurring and non-terminating decimal),

so state your conclusion.

13is a surd.

c 1 Evaluate1

16

. c1

16

1

4

=

2 The answer is rational (a fraction); state

your conclusion.

1

16is not a surd.

d 1 Evaluate 173 . d 173 =2.571 281 590 66 . . .

2 The answer is irrational (a non-

terminating and non-recurring decimal),

so state your conclusion.

173 is a surd.

e 1 Evaluate 634 . e 634 =2.817 313 247 26 . . .

2 The answer is irrational, so classify 634 accordingly.

634 is a surd.

f 1 Evaluate 17283 . f 1728 123 =

2 The answer is rational; state your

conclusion.

17283 is not a surd.

So b, dand eare surds.

Proof that a number is irrational In Mathematics you are required to study a variety of types of proofs. One such method is

called proof by contradiction. This method is so named because the logical argument of the proof is based on an assumption

that leads to contradiction within the proof. Therefore the original assumption must be false.

An irrational number is one that cannot be expressed in the forma

b(where a and bare

integers). The next worked example sets out to prove that 2 is irrational.

Prove that 2 is irrational.

THINK WRITE

1 Assume that 2is rational; that is, it

can be written asa

bin simplest form.

We need to show that aand bhave no

common factors.

Let 2 =a

b, where b0

2 Square both sides of the equation. 22

2=

a

b

3Rearrange the equation to make a

2

thesubject of the formula. a

2

=

2b

2

[1]

WORKED EXAMPLE 3


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4 Ifxis an even number, thenx=2n. \a2is an even number and amust also be even;

that is, ahas a factor of 2.

5 Since ais even it can be written as a=2r. \a=2r

6 Square both sides. a2=4r2 [2]

But a2=2b2 from [1]

7 Equate [1] and [2]. \2b2=4r2

b r2

24

2=

=2r2

\b2 is an even number and bmust also be even;

that is, bhas a factor of 2.

8 Repeat the steps for bas previously done for a. Both aand bhave a common factor of 2.

This contradicts the original assumption that

2 =a

b, where aand bhave no common factor.

\ 2is not rational.\It must be irrational.

The dialogue included in the worked example should be present in all proofs and is an

essential part of the communication that is needed in all your solutions. Note:An irrational number written in surd form gives an exact value of the number; whereas

the same number written in decimal form (for example, to 4 decimal places) gives an

approximate value.

REMEMBER

A number is a surd if:

1. it is an irrational number (equals a non-terminating, non-recurring decimal)

2. it can be written with a radical sign (or square root sign) in its exact form.

Surds

FLUENCY 1 WE 2 Which of the numbers below are surds?

a 81 b 48 c 16 d 1 6. e 0 16. f 11

g3

4h

3

27

3 i 1000 j 1 44. k 4 100 l 2 10+

m 323 n 361 o 1003 p 1253 q 6 6+ r 2p

s 1693 t7

8u 164 v ( )7

2w 333 x 0 0001.

y 325 z 80

EXERCISE

18B

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2 MC The correct statement regarding the set of numbers6

920 54 27 9

3, , , ,

is:

A 273 and 9 are the only rational numbers of the set.

B6

9is the only surd of the set.

C6

9and 20 are the only surds of the set.

D 20 and 54 are the only surds of the set.

E 9 and 20 are the only surds of the set.

3 MC Which of the numbers of the set1

4

1

27

1

821 83

3, , , ,

are surds?

A 21only B1

8

only C1

8

and 83

D1

8and 21only E

1

4and 21only

4 MC Which statement regarding the set of numbers , , , ,1

4912 16 3 1+

is nottrue?

A 12 is a surd. B 12 and 16 are surds.

C pis irrational but not a surd. D 12 and 3 1+ are not rational.

Ep

is not a surd.

5 MC Which statement regarding the set of numbers 6 7 144

167 6 9 2 18 25, , , , ,

is

nottrue?

A144

16when simplified is an integer. B

144

16and 25are not surds.

C 7 6is smaller than 9 2 . D 9 2 is smaller than 6 7.

E 18is a surd.

UNDERSTANDING

6 Complete the following statement by selecting appropriate words, suggested in brackets:

a6 is definitely not a surd, if ais . . . (any multiple of 4; a perfect square and cube).

7 Find the smallest value of m,where mis a positive integer, so that 163 m is not a surd.

REASONING

8 WE 3 Prove that the following numbers are irrational, using a proof by contradiction:

a 3

b 5

c 7

REFLECTIONHow can you be certain that

a

is a surd?


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Operations with surdsSimplifying surds

To simplify a surd means to make a number (or an expression) under the radical sign ( ) as

small as possible. To simplify a surd (if it is possible), it should be rewritten as a product of two factors, one of

which is a perfect square, that is, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. We must always aim to obtain the largest perfect square when simplifying surds so that

there are fewer steps involved in obtaining the answer. For example, 32 could be written

as 4 8 2 8 = ; however, 8 can be further simplified to 2 2 , so 32 2 2 2= ; that is

32 4 2= . If, however, the largest perfect square had been selected and 32 had been

written as 16 2 16 2 4 2 = = , the same answer would be obtained in fewer steps.

Simplify the following surds. Assume that xand yare positive real numbers.

a 384 b 3 405 c -1

8175 d 5 180

3 5x y

THINK WRITE

a 1 Express 384 as a product of two factors

where one factor is the largest possible

perfect square.

a 384 64 6=

2 Express 64 6 as the product of two

surds.

= 64 6

3 Simplify the square root from the perfectsquare (that is, 64 8= ).

= 8 6

b 1 Express 405 as a product of two factors,

one of which is the largest possible

perfect square.

b 3 405 3 81 5=

2 Express 81 5 as a product of two

surds.

= 3 81 5

3 Simplify 81. = 3 9 5

4 Multiply together the whole numbers

outside the square root sign (3 and 9).

= 27 5

c 1 Express 175 as a product of two factors

in which one factor is the largest possible

perfect square.

c = 1

8175

1

825 7

2 Express 25 7 as a product of 2 surds. = 1

825 7

3 Simplify 25. = 1

85 7

4Multiply together the numbers outsidethe square root sign.

=

5

87

18C

WORKED EXAMPLE 4


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d 1 Express each of 180,x3andy5as a

product of two factors where one factor

is the largest possible perfect square.

d 5 180 5 36 53 5 2 4x y x x y y=

2 Separate all perfect squares into one surd

and all other factors into the other surd.

= 5 36 52 4

x y xy

3 Simplify 36 2 4

x y . = 5 6 52

x y xy

4 Multiply together the numbers and the

pronumerals outside the square root sign.

=30 52

xy xy

Addition and subtraction of surds Surds may be added or subtracted only if they are alike.

Examples of likesurds include 7, 3 7and 5 7. Examples of unlikesurds include

11 5 2 13, , and 2 3. In some cases surds will need to be simplified before you decide whether they are like

or unlike, and then addition and subtraction can take place. The concept of adding andsubtracting surds is similar to adding and subtracting like terms in algebra.

Simplify each of the following expressions containing surds. Assume that aand bare positive real

numbers.

a 3 6 17 6 2 6+ b 5 3 2 12 5 2 3 8+ + c1

2100 36 5 4

3 2 2a b ab a a b+

THINK WRITE

a All 3 terms are alike because they contain the

same surd ( )6 .

Simplify.

a 3 6 17 6 2 6 3 17 2 6+ = + ( )

= 18 6

b 1 Simplify surds where possible. b 5 3 2 12 5 2 3 8+ +

= + + 5 3 2 4 3 5 2 3 4 2

= + + 5 3 2 2 3 5 2 3 2 2

= + +5 3 4 3 5 2 6 2

2 Add like terms to obtain the simplified

answer.

= +9 3 2

c 1 Simplify surds where possible. c 1

2100 36 5 4

3 2 2a b ab a a b+

= + 1

210 6 5 2

2 2a a b ab a a b

= + 1

210 6 5 2a b a ab a a b

= + 5 6 10ab a ab a a b

2 Add like terms to obtain the simplified

answer.

= 11 10ab a a b

WORKED EXAMPLE 5


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Multiplication and division of surdsMultiplying surds

To multiply surds, multiply together the expressions under the radical signs. For example,

a b ab = , where aand bare positive real numbers. When multiplying surds it is best to first simplify them (if possible). Once this has been done

and a mixed surd has been obtained, the coefficients are multiplied with each other and then

the surds are multiplied together. For example,

m a n b mn ab =

Multiply the following surds, expressing answers in the simplest form. Assume thatxand yare

positive real numbers.

a 11 7 b 5 3 8 5 c 6 12 2 6 d 15 125 2 2

x y x y

THINK WRITE

a Multiply the surds together, using

a b ab = (that is, multiply expressions under the

square root sign).

Note:This expression cannot be simplified any further.

a 11 7 11 7 =

= 77

b Multiply the coefficients together and then multiply the

surds together.

b 5 3 8 5 5 8 3 5 =

= 40 3 5

= 40 15

c 1 Simplify 12 . c 6 12 2 6 6 4 3 2 6 =

= 6 2 3 2 6

= 12 3 2 6

2 Multiply the coefficients together and multiply

the surds together.

= 24 18

3 Simplify the surd. = 24 9 2

= 24 3 2

= 72 2

d 1 Simplify each of the surds. d 15 125 2 2x y x y

= 15 4 3

4 2 2

x x y x y

= x y x x y2

15 2 3

= x y x x y2

15 2 3

2 Multiply the coefficients together and

the surds together.

= x y x x y2

2 15 3

= 2 453

x y xy

3 Simplify the surd. = 2 9 53

x y xy

= 2 3 53

x y xy

= 6 53

x y xy

WORKED EXAMPLE 6


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When working with surds, we sometimes need to multiply surds by themselves; that is, square

them. Consider the following examples:

( )2 2 2 4 22

= = =

( )5 5 5 25 52

= = =

Observe that squaring a surd produces the number under the radical sign. This is not

surprising, because squaring and taking the square root are inverse operationsand, when

applied together, leave the original unchanged. When a surd is squared, the result is the number (or expression) under the radical sign; that is,

( )a2=a, where ais a positive real number.

Simplify each of the following.

a ( )6 2

b ( )3 5 2

THINK WRITE

a Use ( )a a2= , where a=6. a ( )6 6

2=

b 1 Square 3 and use ( )a a2= to square 5. b ( ) ( )3 5 3 5

2 2 2=

= 9 5

2 Simplify. =45

Dividing surds To divide surds, divide the expressions under the radical signs; that is,

a

b

a

b= , where aand

bare whole numbers. When dividing surds it is best to simplify them (if possible) first. Once this has been done, the

coefficients are divided next and then the surds are divided.

Divide the following surds, expressing answers in the simplest form. Assume that xand yare positive

real numbers.

a55

5

b48

3

c9 88

6 99

d36

25 9 11

xy

x y

THINK WRITE

a 1 Rewrite the fraction, usinga

b

a

b= . a

55

5

55

5=

WORKED EXAMPLE 7

WORKED EXAMPLE 8


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2 Divide the numerator by the denominator

(that is, 55 by 5).= 11

3 Check if the surd can be simplified any

further.

b 1 Rewrite the fraction, usinga

b

a

b= . b

48

3

48

3=

2 Divide 48 by 3. = 16

3 Evaluate 16 . =4

c 1 Rewrite surds, usinga

b

a

b= . c

9 88

6 99

9

6

88

99=

2 Simplify the fraction under the radical

by dividing both numerator and

denominator by 11.

=

9

6

8

9

3 Simplify surds. =

9 2 2

6 3

4 Multiply the whole numbers in the

numerator together and those in the

denominator together.

=

18 2

18

5 Cancel the common factor of 18. = 2

d 1 Simplify each surd. d 36

25

6

59 11 8 10

xy

x y

xy

x x y y

=

=6

5 4 5

xy

x y xy

2 Cancel any common factors in this

case xy.

=

6

5 4 5

x y

Rationalising denominators If the denominator of a fraction is a surd, it can be changed into a rational number. In other

words, it can be rationalised. As discussed earlier in this chapter, squaring a simple surd (that is, multiplying it by itself)

results in a rational number. This fact can be used to rationalise denominators as follows.

a

b

b

b

ab

b = , where

b

b

= 1

If both numerator and denominator of a fraction are multiplied by the surd contained in the

denominator, the denominator becomes a rational number. The fraction takes on a different

appearance, but its numerical value is unchanged, because multiplying the numerator and

denominator by the same number is equivalent to multiplying by 1.


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Express the following in their simplest form with a rational denominator.

a6

13 b

2 12

3 54 c

17 3 14

7

THINK WRITE

a 1 Write the fraction. a 6

13

2 Multiply both the numerator and denominator by the surd

contained in the denominator (in this case 13). This has the

same effect as multiplying the fraction by 1, because13

13

1= .

= 6

13

13

13

=

78

13

b 1 Write the fraction. b 2 12

3 54

2 Simplify the surds. (This avoids dealing with large numbers.) 2 12

3 54

2 4 3

3 9 6=

=

2 2 3

3 3 6

=

4 3

9 6

3 Multiply both the numerator and denominator by 6 .

(This has the same effect as multiplying the fraction by 1,

because6

6

1= .)

Note:We need to multiply only by the surd part of the

denominator (that is, by 6 rather than by 9 6).

= 4 3

9 6

6

6

=

4 18

9 6

4 Simplify 18. =

4 9 2

9 6

=4 3 2

54

=

12 2

54

5 Divide both the numerator and denominator by 6 (cancel down). =2 2

9

c 1 Write the fraction. c 17 3 14

7

2 Multiply both the numerator and denominator by 7. Use

grouping symbols (brackets) to make it clear that the wholenumerator must be multiplied by 7.

=

( )17 3 14

7

7

7

WORKED EXAMPLE 9


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3 Apply the Distributive Law in the numerator.

a(b+c) =ab+ac

=

17 7 3 14 7

7 7

=

119 3 98

7

4 Simplify 98. = 119 3 49 2

7

= 119 3 7 2

7

=

119 21 2

7

Rationalising denominators using conjugate surds The product of pairs of conjugate surds results in a rational number. (Examples of pairs

of conjugate surds include 6 11+ and 6 11 +, a band a b

, 2 5 7and 2 5 7+ .)

This fact is used to rationalise denominators containing a sum or a difference of surds. To rationalise the denominator that contains a sum or a difference of surds, multiply both

numerator and denominator by the conjugate of the denominator.

Two examples are given below:

1. To rationalise the denominator of the fraction1

a b+

, multiply it bya b

a b

.

2. To rationalise the denominator of the fraction1

a b

, multiply it bya b

a b

+

+

.

A quick way to simplify the denominator is to use the difference of two squares identity:

( )( ) ( ) ( )a b a b a b + = 2 2

=a-b

Rationalise the denominator and simplify the following.

a1

4 3

b6 3 2

3 3

+

+

THINK WRITE

a 1 Write down the fraction. a 1

4 3

2 Multiply the numerator and denominator

by the conjugate of the denominator.

(Note that

( )

( )

4 3

4 3 1

+

+= .)

=

+

+

1

4 3

4 3

4 3( )

( )

( )

WORKED EXAMPLE 10


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3 Apply the Distributive Law in the

numerator and the difference of

two squares identity in the denominator.

=+

4 3

4 32 2

( ) ( )

4 Simplify.=

+

4 3

16 3

=

+4 3

13

b 1 Write down the fraction. b 6 3 2

3 3

+

+

2 Multiply the numerator and denominator

by the conjugate of the denominator.

(Note that( )

( )

3 3

3 31

= .)

=+

+

( )

( )

( )

( )

6 3 2

3 3

3 3

3 3

3 Multiply the expressions in grouping

symbols in the numerator, and apply the

difference of two squares identity in the

denominator.

= + + +

6 3 6 3 3 2 3 3 2 3

3 32 2

( ) ( )

4 Simplify.

= +

3 6 18 9 2 3 6

9 3

=

+18 9 2

6

=

+9 2 9 2

6

=

+3 2 9 2

6

=

6 2

6

= 2

REMEMBER

1. To simplify a surd means to make a number (or an expression) under the radical sign as

small as possible.

2. To simplify a surd, write it as a product of two factors, one of which is the largest

possible perfect square.

3. Only like surds may be added and subtracted.

4. Surds may need to be simplified before adding and subtracting.

5. When multiplying surds, simplify the surd if possible and then apply the following

rules:

(a) a b ab =

(b)m a n b mn ab = , where aand bare positive real numbers.


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Evaluate each of the following without using a calculator.

a 9

1

2 b 64

1

3

THINK WRITE

a 1 Write 9

1

2 as 9. a 9 9

1

2=

2 Evaluate. =3

b 1 Write 64

1

3as 643 . b 64 64

1

3 3=

2 Evaluate. =4

WORKED EXAMPLE 12

Use a calculator to find the value of the following, correct to 1 decimal place.

a 10

1

4 b 200

1

5

THINK WRITE

a Use a calculator to produce the answer. a 10

1

4 =1.778 279 41

1.8

b Use a calculator to produce the answer. b 200

1

5 =2.885 399 812

2.9

Consider the expression ( )am

n

1

. Using our work so far on fractional indices, we can say

( )a am

n mn

1

= .

We can also say ( )a am n

m

n

1

= using the index laws.

We can therefore conclude that a a

m

n mn

= . Such expressions can be evaluated on a calculator either by using the index function, which is

usually either ^ orxyand entering the fractional index, or by separating the two functions for

power and root.

WORKED EXAMPLE 13

Evaluate 3

2

7 , correct to 1 decimal place.

THINK WRITE

Use a calculator to evaluate 3

2

7. 3

2

71 4.

We can also use the index lawa a

1

2=

to convert between expressions that involve fractionalindices and surds.

WORKED EXAMPLE 11


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Write each of the following expressions in simplest surd form.

a 10

1

2 b 5

3

2

THINK WRITE

a Since an index of1

2is equivalent to taking the square root, this

term can be written as the square root of 10.

a 10

1

2= 10

b 1 A power of2means square root of the number cubed. b 5 5

3

2 3=

2 Evaluate 53. = 125

3 Simplify 125. = 5 5

In Year 9 you would have studied the index laws and all of these laws are valid for fractionalindices.

Simplify each of the following.

a m m

1

5

2

5 b ( )a b

2 3

1

6 cx

y

2

3

3

4

1

2

THINK WRITE

a 1 Write the expression. a m m

1

5

2

5

2 Multiply numbers with the same base by adding the indices. = m

3

5

b 1 Write the expression. b ( )a b2 3

1

6

2 Multiply each index inside the grouping symbols

(brackets) by the index on the outside.

= a b

2

6

3

6

3 Simplify the fractions. = a b1

3

1

2

c 1 Write the expression. c x

y

2

3

3

4

1

2

2 Multiply the index in both the numerator and

denominator by the index outside the grouping symbols.

=

x

y

1

3

3

8

WORKED EXAMPLE 14

WORKED EXAMPLE 15


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REMEMBER

1. Fractional indices are those that are expressed as fractions.

2. Numbers with fractional indices can be written as surds, using the following identities:

a an n

1

= a a a

m

n mn n m

= =( )

3. All index laws are applicable to fractional indices.

Fractional indicesFLUENCY

1 WE 11 Evaluate each of the following without using a calculator if necessary.

a 16

1

2 b 25

1

2 c 81

1

2

d 8

1

3 e 27

1

3 f 125

1

3

2 WE 12 Use a calculator to evaluate each of the following, correct to 1 decimal place if necessary.

a 81

1

4 b 16

1

4 c 3

1

3

d 5

1

2 e 7

1

5 f 8

1

9

3 WE 13 Use a calculator to find the value of each of the following, correct to 1 decimal place.

a 12

3

8 b 100

5

9 c 50

2

3

d ( . )0 6

4

5 e3

4

3

4

f4

5

2

3

4 WE 14 Write each of the following expressions in simplest surd form.

a 7

1

2 b 12

1

2 c 72

1

2

d 2

5

2 e 3

3

2 f 10

5

2

5 Write each of the following expressions with a fractional index.

a 5 b 10 c x

d m3

e 2 t f 63

6 WE 15a Simplify each of the following. Leave your answer in index form.

a 4 43

5

1

5 b 2 2

1

8

3

8 c a a

12

13

d x x

3

4

2

5 e 5 2

1

3

1

5m m f

1

24

3

7

2

7b b

g 4 2

2

9y y h2

50 05

3

8

3

4a a . i 5

3

1

2x x


a a b a b

2

3

3

4

1

3

3

4 b x y x y

3

5

2

9

1

5

1

3 c 2 3

1

3

3

5

4

5ab a b

d 6 1

3

3

7

1

4

2

5m m n e x y z x y z3

1

2

1

3

1

6

1

3

1

2 f 2 4

2

5

3

8

1

4

3

4

3

4a b c b c

EXERCISE

18D


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a 3 3

1

2

1

3 b 5 5

2

3

1

4 c 12 12

2

3

2

d a a

6

7

3

7 e x x

3

2

1

4 f

m

m

4

5

5

9

g2

4

3

4

3

5

x

x

h7

21

2

4

3

n

n

i25

20

3

5

1

4

b

b


a x y x y3 24

3

3

5 b a b a b

5

9

2

3

2

5

2

5 c m n n

3

8

4

7

3

83

d 10 5

4

5

2

3

1

4x y x y e5

20

3

4

3

5

1

5

1

4

a b

a b

fp q

p q

7

8

1

4

2

3

1

67


a 2

3

4

3

5( ) b 52

3

1

4( ) c 71

5

6

( )

d ( )a3

1

10 e m

4

9

3

8( ) f 21

2

1

3

b( )

g 4

3

7

14

15

p( ) h xm

n

n

p( ) i 3ma

b

b

c( )UNDERSTANDING

11 WE 15b, c Simplify each of the following.

a a b

1

2

1

3

1

2( ) b ( )a b43

4 c x y

3

5

7

8

2

( )

d 3

1

3

3

5

3

4

1

3

a b c( ) e x y z1

2

2

3

2

5

1

2( ) f ab

3

4

2

3

gm

n

4

5

7

8

2

hb

c

3

5

4

9

2

3

i4

2

7

3

4

1

2x

y

12 MC Note:There may be more than one correct answer.

If a

m

n3

4( ) is equal to a1

4 , thenm andn could not be:

A 1 and 3 B 2 and 6C 3 and 8 D 4 and 9


a a8 b b

93 c m164

d 16 4

x e 8 93

y f 16 8 124

x y

g 27 9 153

m n h 32 5 105 p q i 216 6 183

a b


26/46



REASONING

14 At the start of this chapter we looked at Mannings formula, which is used to calculate the flow

of water in a river during a flood situation. Mannings formula is v R S

n=

2

3

1

2

, whereRis the

hydraulic radius, Sis the slope of the river and nis the roughness coefficient. This formula is

used by meteorologists and civil engineers to analyse potential flood situations. We were asked to find the flow of water in metres per second in the river if

R=8, S=0.0025 and n=0.625.

a Use Mannings formula to find the flow of water in the river.

b To find the volume of water flowing through the river, we multiply the flow rate by the

average cross-sectional area of the river. If the average cross-sectional area is 52 m2, find the

volume of water (in L) flowing through the river each second. (Remember 1 m3=1000 L.)

c If water continues to flow at this rate, what will

be the total amount of water to flow through in

one hour? Justify your answer.

d Use the Internet to find the meaning of the terms

hydraulic radius and roughness coefficient.

Negative indices Consider the following division

2

2

2

3

4

1=

(using the Second Index Law).

Alternatively,2

2

8

16

1

2

3

4 = = .

We can conclude that 2 1

2

1= .

In general form:

a

a

=1 1 and a

a

n

n

=

1.

When using a calculator to evaluate expressions that involve negative indices, we need tofamiliarise ourselves with the keys needed.

REFLECTIONHow will you remember the rule

for fractional indices?

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WORKED EXAMPLE 16

Evaluate each of the following using a calculator.

a 4-1 b 2-4

THINK WRITE

a Use a calculator to evaluate 4-1. a 4-1=0.25

b Use a calculator to evaluate 2-4. b 2-4=0.0625

Consider the index law aa

=1 1

. Now let us look at the case in which ais fractional.

Consider the expressiona

b

1

.

a

b a

b

=1

1

= 1 b

a

=

b

a

We can therefore consider an index of -1 to be a reciprocal function.

Write down the value of each of the following without the use of a calculator.

a2

3

1

b1

5

1

c 11

4

1

THINK WRITE

a To evaluate2

3

1

take the reciprocal of2

3. a

2

3

3

2

1

=

b 1 To evaluate1

5

1

take the reciprocal of1

5. b

1

5

5

1

1

=

2 Write5

1as a whole number. =5

c 1 Write 11

4as an improper fraction. c 1

1

4

5

4

1 1

=

2 Take the reciprocal of5

4. =

4

5

WORKED EXAMPLE 17


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REMEMBER

1. To evaluate an expression that involves negative indices, use the1

xy

or thex-1function.

2. An index of -1 can be considered as a reciprocal function and applying this to fractions

gives us the rulea

b

b

a

=

1

.

Negative indicesFLUENCY

1 WE 16 Evaluate each of the following using a calculator.

a 5-1 b 3-1 c 8-1

d 10-1 e 2-3 f 3-2

g 5-2 h 10-4

2 Find the value of each of the following, correct to 3 significant figures.

a 6-1 b 7-1 c 6-2

d 9-3 e 6-3 f 15-2

g 16-2 h 5-4


a (2.5)-1 b (0.4)-1 c (1.5)-2

d (0.5)-2 e (2.1)-3 f (10.6)-4

g (0.45)-3 h (0.125)-4


a (-3)-1 b (-5)-1 c (-2)-2

d (-4)-2

e (-1.5)-1

f (-2.2)-1

g (-0.6)-1 h (-0.85)-2

5 WE 17 Write down the value of each of the following without the use of a calculator.

a4

5

1

b3

10

1

c7

8

1

d13

20

1

e1

2

1

f1

4

1

g1

8

1

h1

10

1

i 11

2

1

j 21

4

1

k 11

10

1

l 51

2

1

6 Find the value of each of the following, leaving your answer in fraction form if necessary.

a1

2

2

b2

5

2

c2

3

3

d1

4

2

e 11

2

2

f 21

4

2

g 11

3

3

h 21

5

3

7 Find the value of each of the following.

a

2

3

1

b

3

5

1

c

1

4

1

d

1

10

1

e

2

3

2

f

1

5

2

g

1

1

2

1

h

2

3

4

2

EXERCISE

18E


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REASONING

8 Consider the expression 2-n. Explain what happens to

the value of this expression as nincreases.

Logarithms The index, power or exponent in the statementy=axis also known as a logarithm(or log for

short).

Logarithm (or index or power or exponent)

y=ax

Base

This statementy=axcan be written in an alternative form as logay=x, which is read as the

logarithm ofyto the base ais equal tox. These two statements are equivalent.

ax=y logay=x

Index form Logarithmic form

For example, 32=9 can be written as log39 =2. The log form would be read as the

logarithm of 9, to the base of 3, is 2. In both forms, the base is 3 and the logarithm is 2.

Write the following in logarithmic form.

a 104=10 000 b 6x=216

THINK WRITE

a 1 Write the given statement. a 104=10 000

2 Identify the base (10) and the logarithm (4) and write

the equivalent statement in logarithmic form. (Use

ax=ylogay=x, where the base is aand the log isx.)

log1010 000 =4

b 1 Write the given statement. b 6x=216

2 Identify the base (6) and the logarithm (x) and write

the equivalent statement in logarithmic form.

log6216 =x

Write the following in index form.

a log28 =3 b log255 =1

2

THINK WRITE

a 1 Write the statement. a log28 =3

2 Identify the base (2) and the log (3) and write the

equivalent statement in index form. Remember that thelog is the same as the index.

23=8

REFLECTIONHow can division to used toexplain negative indices?

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WORKED EXAMPLE 18

WORKED EXAMPLE 19


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b 1 Write the statement. b log255 =1

2

2 Identify the base (25) and the log1

2

and write the equivalent statement in index form.

25

1

2 =5

In the previous examples, we found that:

log28 =3 23=8 and log1010 000 =4 10

4=10 000.

We could also write log28 =3 as log223=3 and log1010 000 =4 as log1010

4=4. Can this pattern be used to work out the value of log381? We need to find the power when the

base of 3 is raised to that power to give 81.

WORKED EXAMPLE 20

Evaluate log381.

THINK WRITE

1 Write the log expression. log381

2 Express 81 in index form with a base of 3. =log334

3 Write the value of the logarithm. =4

REMEMBER

1. Logarithm is another name for an index, power or exponent.

For example, in the statement 23=8, the logarithm is 3.

2. The logarithm of a number to any positive base is the index when the number isexpressed as a power of the base.

That is, ax=ylogay=x, where a>0,y >0.

3. One way of evaluating the logarithm of a number is to write the number in index form

to the given base.

That is, logaax=x.

For example, log381 =log334=4.

LogarithmsFLUENCY

1 WE 18 Write the following in logarithmic form.

a 42=16 b 25=32 c 34=81 d 62=36

e 1000 =103 f 25 =52 g 43=x h 5x=125

i 7x=49 j p4=16 k 9 3

1

2= l 0.1 =10-1

m 2 8

1

3= n 2

1

2

1= o a0=1 p 4 8

3

2=

2 MC The statement w=htis equivalent to:

A w=

logth B h=

logtwC t=logwh D t=loghw

EXERCISE

18F


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3 WE 19 Write the following in index form.

a log216 =4 b log327 =3 c log101 000 000 =6

d log5125 =3 e log164 =1

2f log464 =x

g1

2=log497 h log3x=5 i log819 =

1

2

j log100.01 =-2 k log88 =1 l log644 =1

3

4 MC The statement q=logrpis equivalent to:

A q=rp B p=rq

C r=pq D r=qp

5 WE 20 Evaluate the following logarithms.

a log216 b log416

c log11121 d log10100 000

e log3243 f log2128

g log51 h log93

i log31

3 j log66

k log101

100

l log1255

6 Write the value of each of the following.

a log101 b log1010

c log10100 d log101000

e log1010 000 f log10100 000

UNDERSTANDING

7 Use your results to question 6to answer the following.

a Between which two whole numbers would log107 lie?

b Between which two whole numbers would log104600 lie?

c Between which two whole numbers would log1085 lie?

d Between which two whole numbers would log1012 750 lie?

e Between which two whole numbers would log10110 lie?

f Between which two whole numbers would log1081 000 lie?

REASONING

8 a If log10g=k, find the value of log10g2. Justify your answer.

b If logxy =2, find the value of logyx. Justify your answer.

c By referring to the equivalent index statement, explainwhyxmust be a positive number given log4x=y, for all

values ofy.

Logarithm laws From previous work, you will be familiar with the index laws.

1. aman=am+n 2.a

a

a

m

n

m n

=

3. (am)n=amn

4. a0 =1 5. a1 =a 6. aa

=1 1

We can use these index laws to produce equivalent logarithm laws.

REFLECTIONHow are indices andlogarithms related?

18G


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Law 1 Ifx=amandy=an, then logax=mand logay=n(equivalent log form).

Now xy=aman

or xy=am+n (First Index Law).

So loga(xy) =m+n (equivalent log form)

or loga(xy) =logax+logay (substituting for mand n).

logax+logay=loga(xy) This means that the sum of two logarithms with the same base is equal to the logarithm of the

product of the numbers.

WORKED EXAMPLE 21

Evaluate log1020 +log105.

THINK WRITE

1 Since the same base of 10 is used in each log

term, use

logax+logay=loga(xy) and simplify.

log1020 +log105 =log10(20 5)

=log10100

2 Evaluate. (Remember that 100 =102.) =2

Law 2 Ifx=amandy=an, then logax=mand logay=n(equivalent log form).

Nowx

y

a

a

m

n=

orx

y

am n

= (Second Index Law).

So logax

ym n

= (equivalent log form)

or log log loga a ax

yx y

= (substituting for mand n).

logax-logay=logax

y

This means that the difference of two logarithms with the same base is equal to the logarithm

of the quotient of the numbers.

WORKED EXAMPLE 22

Evaluate log420 -log45.

THINK WRITE

1 Since the same base of 4 is used in each

log term, use log log loga a a

x yx

y =

and

simplify.

log420 -log45 =log420

5

=log44

2Evaluate. (Remember that 4

=

4

1

.) =

1


33/46



WORKED EXAMPLE 23

Evaluate log535 +log515 -log521.

THINK WRITE

1 Since the first two log terms are being added,

use loga

x+loga

y=loga

(xy) and simplify.

log535 +log515 -log521

=log5

(35 15) -log5

21

=log5525 -log521

2 To find the difference between the two

remaining log terms, use

log log loga a a

x yx

y =

and simplify.

=

log5

525

21

=log525

3 Evaluate. (Remember that 25 =52.) =2

Once you have gained confidence in using the first two laws, you can reduce the number of

steps of working by combining the application of the laws. In Worked example 23, we could

write:

log log log log5 5 5 535 15 21 35 15

21+ =

=log525

=2

Law 3 Ifx=am, then logax=m(equivalent log form).

Now xn=(am)n

or xn=amn (Third Index Law).So logax

n=mn (equivalent log form)

or logaxn=(logax) n (substituting for m)

or logaxn=nlogax.

logaxn=nlogax

This means that the logarithm of a number raised to a power is equal to the product of the

power and the logarithm of the number.

WORKED EXAMPLE 24

Evaluate 2 log63 +log64.

THINK WRITE

1 The first log term is not in the required form

to use the log law relating to sums. Use

logaxn=n logaxto rewrite the first term in

preparation for applying the first log law.

2 log63 +log64 =log632+log64

=log69 +log64

2 Use logax+logay=loga(xy) to simplify the

two log terms to one.

=log6(9 4)

=log636

3Evaluate. (Remember that 36

=

6

2

.) =

2


34/46



Law 4 As a0 =1 (Fourth Index Law),

loga1 =0 (equivalent log form).

loga1=0

This means that the logarithm of 1 with any base is equal to 0.

Law 5 As a1=a (Fifth Index Law),

logaa=1 (equivalent log form).

logaa=1

This means that the logarithm of any number awith base ais equal to 1.

Law 6 Now log log

a a

x

x1 1

= (Sixth Index Law)

or log loga ax

x1

1

= (using the fourth log law)

or log loga a

x

x1

= .

log 1

loga a

x

x

==

Law 7 Now logaa

x=xlogaa (using the third log law)

or logaax=x1 (using the fifth log law)

or logaax=x.

logaax=x

REMEMBER

The index laws can be used to produce the following logarithm laws.

1. logax+logay=loga(xy) 2. log log loga a ax yx

y =

3. logaxn=n logax 4. loga1 =0

5. logaa=1 6. log loga ax

x1

=

7. logaax=x

Logarithm lawsFLUENCY

1 Use a calculator to evaluate the following, correct to 5 decimal places.

a log1050 b log1025 c log105 d log102

2 Use your answers to question 1to show that each of the following statements is true.

a log1025+

log102=

log1050 b log1050-

log102=

log10 25c log1025 =2 log105 d log1050 -log1025 -log102 =log10 1

EXERCISE

18G


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3 WE 21 Evaluate the following.

a log63 +log62 b log48 +log48

c log1025 +log104 d log8 32 +log816

e log6108 +log612 f log142 +log147


a log220 -log25 b log354 -log32

c log424 -log46 d log10 30 000 -log103

e log6648 -log63 f log2224 -log27


a log327 +log32 -log36 b log424 -log42 -log46

c log678 -log613 +log61 d log2120 -log23 -log25

6 Evaluate 2 log48.


a 2 log105 +log104 b log3648 -3 log32

c 4 log510 -log580 d log250 +1

2log216 -2 log25

8 Evaluate the following.

a log88 b log51 c log21

2

d log4 45 e log66

-2 f log2020

g log2 1 h log31

9

i log4

1

2

j log5 5 k log31

3

l log2 8 2

UNDERSTANDING

9 Use the logarithm laws to simplify each of the following.

a loga5 +loga8 b loga12 +loga3 -loga2

c 4 logx2 +logx3 d logx100 -2 logx5

e 3 logax-logax2 f 5 logaa-logaa

4

g logx6 -logx6x h logaa7+loga1

i logp

p j logk

k k

k 6 1

loga

a

l loga

a

13

10 MC Note:There may be more than one correct answer.

a The equationy=10x

is equivalent to:A x=10y B x=log10y

C x=logx10 D x=logy10

b The equationy=104xis equivalent to:

A x y=log10 4 B x y=log104

C xy

= 10

1

4 D x y=1

4 10log

c The equationy=103xis equivalent to:

A x y=1

3 10log B x y=log10

1

3

C x=log10y 3 D x=10y 3


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d The equationy=manxis equivalent to:

A xnamy

=

1B x

m

ya

n

=

log

C xn

y ma a

=

1(log log ) D x

n

y

ma

=

1log

11 Simplify, and evaluate where possible, each of the following without a calculator.a log28 +log210 b log37 +log315 c log1020 +log105

d log68 +log67 e log220 -log25 f log336 -log312

g log5100 -log58 h log21

3+log29 i log425 +log4

1

5

j log105 -log1020 k log34

5-log3

1

5l log29 +log24 -log212

m log38 -log32 +log35 n log424 -log42 -log46

12 MC a The expression log10xyis equal to:

A log10xlog10y B log10x-log10y

C log10x+

log10y D ylog10xb The expression log10xyis equal to:

A xlog10y B ylog10x

C 10 logxy D log10x+log10y

c The expression1

3log2 64 +log210 is equal to:

A log240 B log2 80

C log264

10D 1

REASONING

13 For each of the following, write the possible strategy you intend to use.a Evaluate (log381)(log327).

b Evaluatelog

log

a

a

81

3.

c Evaluate 5 57log.

In each case, explain how you obtained your final answer.

Solving equations The equation logay=xis an example of a general logarithmic equation. Laws of logarithms

and indices are used to solve these equations.

Solve for xin the following equations.

a log2x=3 b log6x=-2

c log3x4=-16 d log5(x-1) =2

THINK WRITE

a 1 Write the equation. a log2x=3

2 Rewrite using ax=ylogay=x. 23=x

3 Rearrange and simplify. x=8

REFLECTIONWhat technique will you use

to remember the log laws?

18H

WORKED EXAMPLE 25


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b 1 Write the equation. b log6x=-2

2 Rewrite using ax=ylogay=x. 6-2=x

3 Rearrange and simplify. x=1

62

=

1

36

c 1 Write the equation. c log3x4=-16

2 Rewrite using logaxn=nlogax. 4 log3x=-16

3 Divide both sides by 4. log3x=-4

4 Rewrite using ax=ylogay=x. 3-4=x

5 Rearrange and simplify. x =1

34

=

1

81

d 1 Write the equation. d log5(x-1) =2

2 Rewrite using ax=ylogay=x. 52=x-1

3 Solve forx. x-1 =25

x=26

WORKED EXAMPLE 26

Solve for xin logx25 =2, given that x> 0.

THINK WRITE

1 Write the equation. logx25 =2

2 Rewrite using ax=ylogay=x. x2=25

3 Solve forx.

Note:x=-5 is rejected as a solution

becausex>0.

x=5 (becausex>0)

Solve for xin the following.

a log216 =x b log31

3

=x c log93 =x

THINK WRITE

a 1 Write the equation. a log216 =x

2 Rewrite using ax=ylogay=x. 2x=16

3 Write 16 with base 2. =24

4 Equate the indices. x=4

WORKED EXAMPLE 27


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b 1 Write the equation. b log31

3

= x

2 Rewrite using ax=ylogay=x. 3 1

3

x

=

=

1

3

1

3 Write1

3with base 3. 3x=3-1

4 Equate the indices. x=-1

c 1 Write the equation. c log93 =x

2 Rewrite using ax=ylogay=x. 9x=3

3 Write 9 with base 3. (32)x=3

4 Remove the grouping symbols. 32x=31

5 Equate the indices. 2x=1

6 Solve forx. x =1

2

WORKED EXAMPLE 28

Solve for xin the equation log24 +log2x-log28 =3.

THINK WRITE

1 Write the equation. log24 +log2x-log28 =3

2 Simplify the left-hand side.

Use logax+logay=loga(xy)and

log log loga a a

x yx

y =

.

log24

83

=x

3 Simplify. log22

3x

=

4 Rewrite using ax=ylogay=x. 22

3=

x

5 Solve forx. x=2 23

=2 8

=16

When solving an equation like log28 =x, we could rewrite it in index form as 2x=8. This can

be written with the same base of 2 to produce 2x=23. Equating the indices gives us a solution

ofx=3.

Can we do this to solve the equation 2

x

=

7? Consider the method shown in the next workedexample. It involves the use of logarithms and the log10function on a calculator.


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Solve for x, correct to 3 decimal places, if

a 2x=7 b 3-x=0.4

THINK WRITE

a 1 Write the equation. a 2x=7

2 Take log10of both sides. log102x=log107

3 Use the logarithm-of-a-power law to

bring the power,x, to the front of the

logarithmic equation.

xlog102 =log107

4 Divide both sides by log102 to getxby

itself.

Therefore, x =log

log

10

10

7

2

5 Use a calculator to evaluate the

logarithms and write the answer correct

to 3 decimal places.

=2.807

b 1 Write the equation. b 3-x=0.4

2 Take log10of both sides. log103-x=log100.4

3 Use the logarithm of a power law to

bring the power,x, to the front of the

logarithmic equation.

-xlog103 =log100.4

4 Divide both sides by log103 to get the -x

by itself.

=x

log .

log

10

10

0 4

3

5 Use a calculator to evaluate the

logarithms and write the answer correctto 3 decimal places.

-x=-0.834

6 Divide both sides by -1 to getxby itself. x=0.834

Therefore, we can state the following rule:

Ifax=b,thenxb

a==

log

log

10

10

.

This rule applies to any base, but since your calculator has base 10, this is the most

commonly used for this solution technique.

REMEMBER

1. In a logarithmic equation the unknown,x, can be:

(a) the number, log2x=5

(b) the base, logx8 =3

(c) the logarithm, log24 =x.

2. The laws of logarithms and indices can be used to solve these equations.

3. If ax=b, then xb

a=

log

log

10

10

.

WORKED EXAMPLE 29


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Solving equationsFLUENCY

1 WE 25 Solve forxin the following.

a log5x=2 b log3x=4 c log2x=-3

d log4x=-2 e log10x2=4 f log2x

3=12

g log3(x+1)=3 h log5(x-2)=3 i log4(2x-3)=0j log10(2x+1)=0 k log2(-x)=-5 l log3(-x)=-2

m log5(1 -x)=4 n log10(5 -2x)=1

2 WE 26 Solve forxin the following, given thatx>0.

a logx9 =2 b logx16 =4 c logx25 =2

3

d logx125 =3

4e logx

1

8

=-3 f logx

1

64

=-2

g logx62=2 h logx4

3=3


a log28 =x b log39 =x

c log51

5

=x d log4

1

16

=x

e log42 =x f log82 =x

g log61 =x h log81 =x

i log12

2 = x j log13

9 = x


a log2x+log24 =log220 b log53 +log5x=log518

c log3x-log32 =log35 d log10x-log104 =log102

e log48 -log4x=log42 f log310 -log3x=log35g log64 +log6x=2 h log2x+log25 =1

i 3 -log10x=log102 j 5 -log48 =log4x

k log2x+log26 -log23 =log210 l log2x+log25 -log210 =log23

m log35 -log3x+log32 =log310 n log54 -log5x+log53 =log56

5 MC a The solution to the equation log 7343 =xis:

A x=2 B x=3 C x=1 D x=0

b If log8x=4, thenxis equal to:

A 4096 B 512 C 64 D 2

c Given that logx3 =1

2,xmust be equal to:

A 3 B 6 C 81 D 9d If log ax=0.7, then log ax

2is equal to:

A 0.49 B 1.4 C 0.35 D 0.837

6 Solve forxin the following equations.

a 2x=128 b 3x=9 c 7 1

49

x

= d 9x=1

e 5x=625 f 64x=8 g 6 6x

= h 2 2 2x =

i 3 1

3

x

= j 4x=8 k 9 3 3x

= l 2 1

4 2

x

=

m 3 27 31x+

= n 2

1

32 2

1x

= o 4

1

8 2

1x+

=

EXERCISE

18H


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UNDERSTANDING

7 WE 29 Solve the following equations, correct to 3 decimal places.

a 2x=11 b 2x=0.6 c 3x=20

d 3x=1.7 e 5x=8 f 0.7x=3

g 0.4x=5 h 3x+2=12 i 7-x=0.2

j 8-x=0.3 k 10-2x=7 l 82 -x=0.75

8 The decibel (dB) scale for measuring loudness, d,is given by the formula d=10 log10(I1012),

whereIis the intensity of sound in watts per

square metre.

a Find the number of decibels of sound if the

intensity is 1.

b Find the number of decibels of sound

produced by a jet engine at a distance of

50 metres if the intensity is 10 watts per

square metre.

c Find the intensity of sound if the sound

level of a pneumatic drill 10 metres away is

90 decibels.

d Find how the value of dchanges if the

intensity is doubled. Give your answer to the

nearest decibel.

e Find how the value of dchanges if the

intensity is 10 times as great.

f By what factor does the intensity of sound

have to be multiplied in order to add

20 decibels to the sound level?

REASONING

9 The Richter scale is used to describe the energy of earthquakes. A formula for the Richter

scale is: R =2

3log10K 0.9, whereRis the Richter scale value for an earthquake that

releases Kkilojoules (kJ) of energy.

a Find the Richter scale value for an earthquake that releases the following amounts of

energy:

i 1000 kJ ii 2000 kJ iii 3000 kJ

iv 10 000 kJ v 100 000 kJ vi 1 000 000 kJ

b Does doubling the energy released double the Richter scale value? Justify your answer.

c Find the energy released by an earthquake of:

i magnitude 4 on the Richter scale

ii magnitude 5 on the Richter scale

iii magnitude 6 on the Richter scale.

d What is the effect (on the amount of energy released) of

increasing the Richter scale value by 1?

e Why is an earthquake measuring 8 on the Richter scale

so much more devastating than one that measures 5?

eBookpluseBookplus

Digital doc

WorkSHEET 18.4

doc-6754

REFLECTIONTables of logarithms were used in classrooms before calculators were used

there. Would using logarithms have any effect on the accuracy of calculations?


42/46



SummaryNumber classification review

Rational numbers (Q) can be expressed in the forma

b, where aand bare whole numbers

and b0. They include whole numbers, fractions and terminating and recurring decimals.

Irrational numbers (I) cannot be expressed in the form a


numbers and b0. They include surds, non-terminating and non-recurring decimals, and

numbers such as pand e. Rational and irrational numbers together constitute the set of real numbers (R).

Surds

A number is a surd if: it is an irrational number (equals a non-terminating, non-recurring decimal) it can be written with a radical sign (or square root sign) in its exact form.

Operations with surds To simplify a surd means to make a number (or an expression) under the radical sign as small

as possible. To simplify a surd, write it as a product of two factors, one of which is the largest possible

perfect square. Only like surds may be added and subtracted. Surds may need to be simplified before adding and subtracting. When multiplying surds, simplify the surd if possible and then apply the following rules:

(a) a b ab =

(b) m a n b mn ab = , where aand bare positive real numbers. When a surd is squared, the result is the number (or the expression) under the radical

sign: ( )a a2= , where ais a positive real number.

When dividing surds, simplify the surd if possible and then apply the following rule:

a b a

b

a

b = =

where aand bare whole numbers, and b0. To rationalise a surd denominator, multiply the numerator and denominator by the surd

contained in the denominator. This has the effect of multiplying the fraction by 1, and thus the

numerical value of the fraction remains unchanged, while the denominator becomes rational:

a

b

a

b

b

b

ab

b

= =

where aand bare whole numbers and b0. To rationalise the denominator containing a sum or a difference of surds, multiply both

the numerator and denominator of the fraction by the conjugate of the denominator. This

eliminates the middle terms and leaves a rational number.

Fractional indices

Fractional indices are those that are expressed as fractions. Numbers with fractional indices can be written as surds, using the following identities:

a an n

1

= a a a

m

n mn n m

= =( )

All index laws are applicable to fractional indices.


43/46



Negative indices

To evaluate an expression that involves negative indices, use the1

xy

or thex-1function.

An index of -1 can be considered as a reciprocal function and applying this to fractions

gives us the rulea

b

b

a

=1

.

Logarithms

Logarithm is another name for an index, power or exponent.

For example, in the statement 23=8, the logarithm is 3. The logarithm of a number to any positive base is the index when the number is expressed as

a power of the base.

That is, ax=ylogay=x, where a>0,y >0. One way of evaluating the logarithm of a number is to write the number in index form to the

given base.

That is, logaax=x.

For example, log381 =log334=4.

Logarithm laws The index laws can be used to produce the following logarithm laws.

1. logax+logay=loga(xy)

2. log log loga a a

x yx

y =

3. logaxn=n logax

4. loga1 =0

5. logaa=1

6. log loga a

x

x1

=

7. logaax=x

Solving equations

In a logarithmic equation the unknown,x, can be:

(a) the number, log2x=5

(b) the base, logx8 =3

(c) the logarithm, log24 =x. The laws of logarithms and indices can be used to solve these equations.

If ax=b, then xb

a=

log

log

10

10

.

MAPPING YOUR UNDERSTANDING

Using terms from the summary, and other terms if you wish, construct a concept map that

illustrates your understanding of the key concepts covered in this chapter. Compare your

concept map with the one that you created in What do you know?on page 589.


44/46


45/46



16 Evaluate each of the following, without using a

calculator. Show all working.

a16 81

6 16

3

4

1

4

1

2

b 125 27

2

3

2

3

1

2

( )

17 Evaluate each of the following, giving your answer

as a fraction.

a 4-1 b 9-1 c 4-2 d 10-3

18 Find the value of each of the following, correct to

3 significant figures.

a 12-1 b 7-2

c (1.25)-1 d (0.2)-4

19 Write down the value of each of the following.

a2

3

1

b7

10

1

c1

5

1

d 31

4

1

20 MC The expression 250 may be simplified to:

A 25 10 B 5 10

C 10 5 D 5 50

21 MC When expressed in its simplest form,

2 98 3 72 is equal to:

A 4 2 B -4

C 2 4 D 4 2

22 MC When expressed in its simplest form,8

32

3x

is

equal to:

Ax x

2B

x3

4

Cx3

2D

x x

4

23 Find the value of the following, giving your answer

in fraction form.

a 25

1

b 23

2

24 Find the value of each of the following, leaving

your answer in fraction form.

a 2-1 b 3-2

c 4-3 d1

2

1

25 Evaluate the following.

a log1218 +log128

b log460 -log415

c log99

8

d 2 log36 -log34

26 Use the logarithm laws to simplify each of the

following.

a loga16 +loga3 -loga2

b logx

x x

c 4 logax-logax2

d 5 1

logx

x

27 Solve forxin the following, given thatx>0.

a log2x=9 b log5x=-2

c logx25 =2 d logx26=6

e log3729 =x f log71 =x

28 Solve forxin the following.

a log54 +log5x=log524

b log3x-log35 =log37

29 Solve forxin the following equations.

a 6 1

36

x

= b 7 1

7

x

=

c 2 8 21x+ =

30 Solve forxin the following equations, correct to

3 decimal places.

a 2x=25 b 0.6x=7

c 9-x=0.84

PROBLEM SOLVING

1 Answer the following. Explain how you reached

your answer.

a What is the hundreds digit in 333

?b What is the ones digit in 6704?

c What is the thousands digit in 91000?

2 a Plot a graph ofy=4xby first producing a table

of values. Label they-intercept and the equation

of any asymptotes.

b Draw the liney=xon the same set of axes.

c Use the property of inverse graphs to draw the

graph ofy=log4x. Label any intercepts and

the equation of any asymptotes.

d Use a graphics calculator or graphing software

to check your graphs.

eBookpluseBookplus

Interactivities

Test Yourself Chapter 18

int-2873

Word search Chapter 18

int-2871

Crossword Chapter 18

int-2872


46/46


eBookpluseBookplus ACTIVITIES

Are you ready?

Digital docs (page 590) SkillSHEET 18.1 (doc-5354): Identifying surds SkillSHEET 18.2 (doc-5355): Simplifying surds

SkillSHEET 18.3 (doc-5356): Adding andsubtracting surds

SkillSHEET 18.4 (doc-5357): Multiplying anddividing surds

SkillSHEET 18.5 (doc-5358): Evaluating numbers inindex form

SkillSHEET 18.6 (doc-5359): Using the index laws

18A Number classification review

Interactivity

Classifying numbers (int-2792) (page 591)

18B Surds

Digital doc

SkillSHEET 18.1 (doc-5354): Identifying surds(page 597)

18C Oprations with surds

Digital docs (pages 6079) SkillSHEET 18.2 (doc-5355): Simplifying surds SkillSHEET 18.3 (doc-5356): Adding and

subtracting surds SkillSHEET 18.4 (doc-5357): Multiplying and

dividing surds SkillSHEET 18.7 (doc-5360): Rationalising

denominators SkillSHEET 18.8 (doc-5361): Conjugate pairs SkillSHEET 18.9 (doc-5362): Applying the

difference of two squares rule to surds WorkSHEET 18.1 (doc-5363): Real numbers I

18D Fractional indices

Digital doc

WorkSHEET 18.2 (doc-5364): Real numbers II(page 614)

18E Negative indices

Digital doc

WorkSHEET 18.3 (doc-5365): Real numbers III(page 617)

18H Solving equations

Digital doc

WorkSHEET 18.4 (doc-6754): Real numbers IV(page 629)

Chapter review

Interactivities (page 633) Test yourself Chapter 18 (int-2873): Take the end-of-

chapter test to test your progress Word search Chapter 18 (int-2871): an interactive

word search involving words associated with thischapter

Crossword Chapter 18 (int-2872): an interactivecrossword using the definitions associated with thechapter

To access eBookPLUS ac tivit ies, log on to

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