MATHS CHAPTER 2 YEAR 10 PDF

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2

Chapter

Indices and surds

What you will learn 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K

Rational numbers and irrational numbers Adding and subtracting surds Multiplying and dividing surds Binomial products Rationalising the denominator Review of index laws REVISION Negative indices REVISION Scientifi c notation REVISION Fractional indices Exponential equations Exponential growth and decay FRINGE

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nSW Syllabus for the australian Curriculum St r and: num ber an d a lge bra Subst r ands: i nDi CES SuR DS anD in DiCES

Outcomes A student operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases. (MA5.1–5NA)

investment returns Does an average investment return of 15% sound good to you? Given this return is compounded annually, a $10 000 investment would grow to over $40 000 after 10 years. This is calculated by multiplying the investment total by 1.15 (to return the original amount plus the 15%) for each year. Using indices, the total investment value after n years would be given by Value = 10 000 × 1.15n. This is an example of an exponential relation that uses indices to link variables. We can use such a rule to introduce the set of special numbers called surds. If, for example, you wanted to calculate the average investment return that delivers $100 000 after 10 years from a $10 000 investment, then you would need to calculate x in 10  100 000 = 10 000 ×  1+ x  . This gives  100  10 x = ( 10 − 1) × 100 ≈ 25.9, representing an annual return of 25.9%. Indices and surds are commonplace in the world of numbers, especially where money is involved!

© David Greenwood et al. 2014 ISBN 978-1-107-67670-1 Photocopying is restricted under law and this material must not be transferred to another party.

A student applies index laws to operate with algebraic expressions involving integer indices. (MA5.2–7NA) A student performs operations with surds and indices. (MA5.3–6NA)

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Chapter 2 Indices and surds

pre-test

70

1 Evaluate the following. a 52 b (-5)2 e 33

f

i

j

16

3-2 169

c

-52

d -(-5)2

g

2-4

h -2 × 2-2

k

3

l

8

2 Write the following in index form. a 2×2×2×2×2 b 3×3×4×4×4 3 Write the following in expanded form. a 53 b 42 31

c

64

c 2×a×a×b

m4

4 Simplify the following, using index laws. a y4 × y3 b b 8 ÷ b5 d t0 e 2d 3e × 5d 4e2 4 3 h 5(mn 7)0 g (2gh )

3

d 7x3y 2 c (a3)5 f 6s 3t ÷ (4st) i 2x ÷ (2x)

5 State whether the following are rational (fractions) or irrational numbers. . c π a 7 b 3.33 3 5 d 4.873 e f 7 6 Simplify by collecting like terms. a 7x - 11x + 5x b 7a - 4b - 3a + 2b c -3ab - 7ba + a 7 Expand the following. a 3(x + 4) b -2(x + 3)

c

2(-x - 5)

d -4(x − 7)

8 Expand and simplify. a (x + 3)(x + 1) b (x − 1)(x + 5)

c

(2x + 3)(x − 4)

d (x − 2)2

9 A sum of $2000 is deposited into an account paying interest at 5% p.a., compounded annually. The investment balance ($A) is given by A = 2000(1.05)t, where t is the time in years. a Find the investment balance, correct to the nearest dollar, after: i 2 years ii 10 years b How many whole years does it take for the investment to at least double in value. Use a trial-and-error approach.

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number and algebra

2A Rational numbers and irrational numbers You will recall that when using Pythagoras’ theorem to find unknown lengths in right-angled triangles, many answers expressed in exact form are surds. The length of the hypotenuse in this triangle, for example, is 5 , which is a surd. A surd is a number that uses a root sign ( ), sometimes called a radical sign.

Stage

√5

1

2

a They are irrational numbers, meaning that they cannot be expressed as a fraction in the form , where a b and b are integers and b ≠ 0. Surds, together with other irrational numbers such as pi (π), and all rational numbers (fractions) make up the entire set of real numbers, which can be illustrated as a point on a number line. –3 (rational) √2 (irrational) π (irrational) –12 (rational)

–3

–2

–1

0

1

2

3

5.3# 5.3 5.3§ 5.2 5.2◊ 5.1 4

4

let’s start: Constructing surds Someone asks you: ‘How do you construct a line 10 cm long?’ Use these steps to construct a line 10 cm long.

C 1 cm A

3 cm

B

wish to use a set square or pair of compasses.

■

■

All real numbers can be located as a point on a number line. Real numbers include: – rational numbers (i.e. numbers that can be expressed as fractions) . 3 4 For example: , − , − 3, 1.6, 2.7, 0.19 7 39 The decimal representation of a rational number is either a terminating or recurring decimal. – irrational numbers (i.e. numbers that cannot be expressed as fractions) For example: 3, -2 7 , 12 - 1, π, 2π - 3 The decimal representation of an irrational number is an infinite non-recurring decimal. Surds are irrational numbers that use a root sign ( ). – For example: 2, 5 11 , - 200 , 1 + 5 – These numbers are not surds: 4 (= 2),

3

Key ideas

Use this idea to construct line segments with the following lengths. You may need more than one triangle for parts d to f . 2 17 20 a b c d 3 e 6 f 22

125 (= 5), − 4 16 (= − 2)

© David Greenwood et al. 2014 ISBN 978-1-107-67670-1 Photocopying is restricted under law and this material must not be transferred to another party.

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Key ideas

Chapter 2 Indices and surds

■

The nth root of a number x is written n x . – If n x = y then y n = x. For example: 5 32 = 2 since 25 = 32.

■

The following rules apply to surds. 2

– ( x )2 = x and x = x when x ≥ 0.

■

■

–

xy =

x × y when x ≥ 0 and y ≥ 0.

–

x = y

x

when x ≥ 0 and y > 0.

y

When a factor of a number is a perfect square we call that factor a square factor. Examples of perfect squares are: 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . When simplifying surds, look for square factors of the number under the root sign, then use a × b = a × b.

Example 1 Defining and locating surds Express each number as a decimal and decide if they are rational or irrational. Then locate all the numbers on the same number line.

S ol u t ion

Exp l an at ion Use a calculator to express as a decimal.

a - 3 = -1.732050807. . . - 3 is irrational.

The decimal does not terminate and there is no recurring pattern.

137 = 1.37 100 137% is rational.

137% is a fraction and can be expressed using a terminating decimal.

. . 3 = 0.428571 or 0.428571 7 3 is rational. 7

3 is an infinitely recurring decimal. 7

b 137% =

c

c 3 7

b 137%

a - 3

Use the decimal equivalents to locate each number on the real number line. 3 7

–√3 –2

–1

0

1.37 1

© David Greenwood et al. 2014 ISBN 978-1-107-67670-1 Photocopying is restricted under law and this material must not be transferred to another party.

2

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number and algebra

73

Example 2 Converting recurring decimals to fractions Convert the following decimals to fractions. . .. a 0.2 b 0.23 S ol u t ion . a 0.2 = 0.22222 . . . Let x = 0.22222 . . . ∴ 10x = 2.22222 . . . 2 – 1 gives 9x = 2 2 ∴x= 9

. c 0.23 Exp l an at ion

There is only one digit that repeats, so multiply by 10. Subtract 1 from 2 .

1 2

Divide both sides by 9. Note: Some calculators are able to do this. Fill your screen with 0.22222 . . . and then press = .

.. 0.2 3 = 0.232323 . . . Let x = 0.232323 . . . ∴ 100x = 23.232323 . . . 2 – 1 gives 99x = 23 23 ∴ x= 99 . c 0.23 = 0.2333333 . . . Let x = 0.2333333 . . . ∴ 10x = 2.3333333 . . . 2 – 1 gives 9x = 2.1 2.1 x= 9 21 x= 90 b

In this example, two digits repeat, so multiply by 100.

1 2

Divide both sides by 99. Use a calculator to check your answer.

Only one digit repeats, so multiply by 10.

1 2

Multiply the numerator and denominator by 10. Use a calculator to check your answer.

Example 3 Simplifying surds Simplify the following. a

b

32

3 200

c

5 40 6

d

75 9

S ol u t ion

Exp l an at ion

a

When simplifying, choose the highest square factor of 32 (i.e. 16 rather than 4), as there is less work to do to arrive at the same answer. Compare with 32 = 4 × 8 = 2 8 = 2 4 × 2 = 4 2

32 = 16 × 2 = 16 × 2 =4 2

b

3 200 = 3 100 × 2 = 3 × 100 × 2 = 3 × 10 × 2 = 30 2

Select the appropriate factors of 200 by finding its highest square factor: 100. Use x × y = x × y and simplify.

© David Greenwood et al. 2014 ISBN 978-1-107-67670-1 Photocopying is restricted under law and this material must not be transferred to another party.

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74

Chapter 2 Indices and surds

c

5 40 5 4 ×10 = 6 6 5 × 4 × 10 = 6 5 10 10 = 63 =

d

75 = 9

Select the appropriate factors of 40. The highest square factor is 4.

Cancel and simplify.

5 10 3 75

Use

9 25 × 3

=

= 9

5 3 3

x = y

x

.

y

Then select the factors of 75 that include a square number and simplify.

Example 4 Expressing as a single square root of a positive integer Express these surds as a square root of a positive integer. a 2 5

b 7 2

S ol u t ion

Exp l an at ion

a 2 5= 4× 5

Write 2 as 4 and then combine the two surds

= 20 b 7 2 = 49 × 2

using

x× y =

xy .

Write 7 as 49 and combine.

= 98

Exercise 2A

U

F PS

Y

C R

LL

T MA

1 Choose the correct word from the words given in italic to make the sentence true. a A number that cannot be expressed as a fraction is a rational/irrational number. b A surd is an irrational number that uses a root/square symbol. c The decimal representation of a surd is a terminating/recurring/non-recurring decimal. d 25 is a surd/rational number.

R K IN G WO

HE M A TI C A

2 Write down the highest square factor of these numbers. For example, the highest square factor of 45 is 9. a 20 b 18 c 125 d 24 e 48 f 96 g 72 h 108

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number and algebra

R K IN G WO

5 Decide if these numbers are surds. a 7 b 2 11 3 9 f −5 3 2 2 6 Simplify the following surds. e

Example 3a

c

2 25

d -5 144

g

1- 3

h 2 1 + 4

a

12

b

45

c

24

d

48

e

75

f

500

g

98

h

90

i

128

j

360

k

162

l

80

a 2 18

b 3 20

c

4 48

e 3 98

f

4 125

g

45 3

h

28 2

54 12

k

80 20

l

99 18

o

2 98 7

p

3 68 21

s

2 108 18

t

3 147 14

U T MA

Example 2

3 Express each number as a decimal and decide if it is rational or irrational. Then locate all the numbers on the same number line. 2 a 5 b 18% c d -124% 5 5 g 2 3 h π f − 2 e 1 7 4 Convert these recurring decimals to fractions. Use your calculator to check your answers. . .. . . b 0.18 c 0.18 d 0.416 a 0.8

F

C R

HE

PS

L LY

Example 1

A M A TI C

7 Simplify the following. Example 3b,c

i

24 4

j

m

3 44 2

n

5 200 25

6 75 4 150 r 5 20 8 Simplify the following. q

Example 3d

Example 4

d 2 63

a

8 9

b

12 49

c

18 25

d

11 25

e

10 9

f

21 144

g

26 32

h

28 50

i

15 27

j

27 4

k

45 72

l

56 76

9 Express these surds as a square root of a positive integer. a 2 3

b 4 2

c

5 2

d 3 3

e 3 5

f

6 3

g

8 2

h 10 7

i

j

5 5

k

7 5

l

9 10

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11 3 Cambridge University Press

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2A

Chapter 2 Indices and surds

R KING WO

U

b

675

1183

c

d

1805

2883

HE

Y

PS

R

LL

a

F

C

T MA

10 Simplify by searching for the highest square factor.

M A T I CA

11 Determine the exact side length, in simplest form, of a square with the given area. a 32 m2 b 120 cm2 c 240 mm2 12 Determine the exact radius and diameter of a circle, in simplest form, with the given area. a 24π cm2 b 54π m2 c 128π m2 13 Use Pythagoras’ theorem to find the unknown length in these triangles, in simplest form. a

b

6m

2 cm 3m 4 cm c

12 mm

d

√7 m

1 mm 2m e

f 4 cm 10 cm

3 mm

WO

R KI N G

U

F

C R

HE

T

72 =

MA

14 Ricky uses the following working to simplify 72. Show how Ricky could have simplified 72 using fewer steps.

PS

L LY

√20 mm

M A TI C A

9× 8

=3 8 = 3 4× 2 = 3× 2 × 2 =6 2 15 a List all the factors of 450 that are perfect squares. b Now simplify 450 using the highest of these factors. 16 Use Pythagoras’ theorem to construct a line segment with the given lengths. You can use only a ruler and a set square or pair of compasses. Do not use a calculator. a

10 cm b 29 cm c 6 cm . 17 Is 0.9 = 1? Search the internet and write an explanation.

© David Greenwood et al. 2014 ISBN 978-1-107-67670-1 Photocopying is restricted under law and this material must not be transferred to another party.

d

22 cm

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number and algebra

77

Enrichment: proving 2 is irrational 18 We will prove that 2 is irrational by the method called ‘proof by contradiction’. Your job is to follow and understand the proof, then copy it out and try explaining it to a friend or teacher. a Before we start, we first need to show that if a perfect square a2 is even then a is even. We do this by showing that if a is even then a2 is even and if a is odd then a2 is odd. If a is even then a = 2k, where k is an integer. If a is odd then a = 2k + 1, where k is an integer. So a2 = (2k)2 = 4k2 = 2 × 2k2, which must be even. ∴ If a2 is even then a is even.

So a2 = (2k + 1)2 = 4k2 + 4k + 1 = 2 × (2k2 + 2k) + 1, which must be odd.

b Now, to prove 2 is irrational let’s suppose that 2 is instead rational and can be written in the a form in simplest form, where a and b are integers (b ≠ 0) and at least one of a or b is odd. b ∴ 2=

a b

a2 b2 2 a = 2b2

So 2 =

(squaring both sides)

∴ a2 is even and, from part a above, a must be even. If a is even then a = 2k, where k is an integer. ∴ If a2 = 2b2 Then (2k)2 = 2b2 4k2 = 2b2 2k2 = b2 2 ∴ b is even and therefore b is even. This is a contradiction because at least one of a or b must be odd. Therefore, the assumption that a 2 can be written in the form must be incorrect and so 2 is irrational. b

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Chapter 2 Indices and surds

2B adding and subtracting surds

Stage

We can apply our knowledge of like terms in algebra to help simplify expressions involving the addition and subtraction of surds. Recall that 7x and 3x are like terms, so 7x + 3x = 10x. The pronumeral x represents any number. When x = 5, then 7 × 5 + 3 × 5 = 10 × 5 and when x = 2 , then 7 2 + 3 2 = 10 2. Multiples of the same surd are called ‘like surds’ and can be collected (i.e. counted) in the same way as we collect like terms in algebra.

5.3# 5.3 5.3§ 5.2 5.2◊ 5.1 4

let’s start: Can 3 2 + 8 be simplified?

Key ideas

To answer this question, first discuss these points.

■

Like surds are multiples of the same surd.

■

For example: 3 , -5 3 , 12 (= 2 3 ), 2 75 (= 10 3 ) Like surds can be added and subtracted. Simplify all surds before attempting to add or subtract them.

■

Example 5 adding and subtracting surds Simplify the following. a 2 3+4 3

b

4 6 +3 2 −3 6 +2 2

S ol u t ion

Exp l an at ion

a 2 3+4 3 =6 3

Collect like surds by adding the coefficients: 2 + 4 = 6.

b

Collect like surds involving 6:

4...