Chapter 01 Surds.pdf

(1)

1

Surds

When applying Pythagoras’ theorem, we have found lengths When applying Pythagoras’ theorem, we have found lengths that cannot be expressed as an exact rational number.

that cannot be expressed as an exact rational number. Pythagoras encountered this when calculating the diagonal Pythagoras encountered this when calculating the diagonal of a square of side length 1 unit. A

of a square of side length 1 unit. A surd surd is a square root is a square root (

(

p

ﬃ

), cube root (), cube root (

p

33

ﬃ

_{), or any type of root whose exact}_{), or any type of root whose exact}

decimal or fraction value cannot be found. decimal or fraction value cannot be found.

(2)

n

_{Chapter outline}

Proﬁciency strands

1-01 Surds and irrational 1-01 Surds and irrational

n

nuummbbeerrss* * U U F F R R CC 1

1--002 2 SSiimmpplliiffyyiinng g ssuurrddss* * U U F F RR 1-03 Adding and subtracting

1-03 Adding and subtracting s

suurrddss* * U U F F RR 1-04 Multiplying and dividing

1-04 Multiplying and dividing s suurrddss* * U U F F RR 1-05 Binomial products 1-05 Binomial products i innvvoollvviinng g ssuurrddss* * U U F F R R CC 1-06 Rationalising the 1-06 Rationalising the d deennoommiinnaattoorr* * U U F F R R CC *STAGE 5.3 *STAGE 5.3

n

_Wordbank

irrational number

irrational number A number such as A number such as pp or or

p

ﬃﬃ

22

ﬃ

that that cannotcannot

be expressed as a fraction

be expressed as a fraction a a

b

rational number

rational number Any number that can be Any number that can be writtewritten in n in thethe

form

form a a

b

b;; where where a a and and b b are integers and are integers and b b

6

6 ¼

¼

00

rationalise the denominator

rationalise the denominator To simplify a fraction To simplify a fraction

involving a surd by making its denominator rational

(that is, not a surd)

real number

real number A number that is either rational or irrational A number that is either rational or irrational

and whose value can be graphed on a

and whose value can be graphed on a numbenumber liner line

simplify a surd

simplify a surd To write a surd To write a surd

p

x x

ﬃﬃ

ﬃ

in its simpin its simplest forlest form som so

that

that x x has no factors that are perfect squares has no factors that are perfect squares

surd

surd A A square root (or other root) whose exacsquare root (or other root) whose exact valuet value

cannot be found cannot be found S S h h u u t t t t e e r r s s t t o o c c k k c c . . o o m m / / t t o o t t o o j j a a n n g g 1 1 9 9 7 7 7 7

(3)

n

_{In this chapter you will:}

•

• (STAG(STAGE 5.3) deﬁne ratioE 5.3) deﬁne rational and irranal and irrationational numberl numbers and perfors and perform operatim operations with surdons with surdss

•

• (STAG(STAGE 5.3) descrE 5.3) describe realibe real, ration, rational and irraal and irrationational numbel numbers and surdrs and surdss •

• (STAG(STAGE 5.3) adE 5.3) add, subd, subtractract, multt, multiply aiply and divnd divide suide surdsrds •

• (STAG(STAGE 5.3) expanE 5.3) expand and simpld and simplify binoify binomial promial productducts involvs involving surding surdss •

• (STAG(STAGE 5.3) ratiE 5.3) rationalionalise the dense the denominominator of expator of expressressions of thions of the forme form a a

p

ﬃﬃ

bb

ﬃ

c

p

ﬃﬃ

d d

ﬃ

SkillCheck

1

1 Simplify Simplify each each expression.expression. a

a (5(5 y y))22 bb (4(4mm))33 cc ((

_

33 x x))22 2

2 Expand Expand each each expression.expression. a

a 5(5( x x

_þ

2)2) bb 4(4( y y

_

3)3) cc 3(13(1

_þ

22ww)) d

d 2(52(5

_

y y)) ee

_

5(2 5(2aa

_þ

3)3) f f k k (1(1

_þ

22k k )) 3

3 Select the squSelect the square numbers are numbers from the followifrom the following list of numng list of numbers.bers. 4

44 4 881 1 225 5 11000 0 775 5 772 2 116 6 550 0 664 4 3322 4

4 Expand Expand and sand simplify implify each eeach expression.xpression. a

a ((mm

_þ

3)(3)(mm

_þ

7)7) bb (( y y

_þ

1)(1)( y y

_

4)4) cc (n(n

_

2)(2)(nn

_

3)3) d d (2(2d d

_þ

3)(13)(1

_þ

33d d )) ee (1(1

_

5 5 p p)(4)(4

_þ

33 p p)) f f (3(3aa

_þ

22 f f )( )(aa

_þ

5 5 f f ) ) g g (( x x

_þ

4)4)22 h h (( y y

_

3)3)22 i i (2(2k k

_þ

1)1)22 j j ((aa

_

5)( 5)(aa

_þ

5) 5) kk ((t t

_þ

7)(7)(t t

_

7)7) ll (3(3mm

_þ

4)(34)(3mm

_

4)4)

1-01

Surds and irrational numbers

A

A surd surd is a square root ( is a square root (

p

ﬃ

), cube root (), cube root (

p

33

ﬃ

_{), or any type of root whose exact decimal or fractional}_{), or any type of root whose exact decimal or fractional}

value cannot be found. As a decimal, its digits run endlessly

value cannot be found. As a decimal, its digits run endlessly without repeating without repeating (like (like pp)) , , so they are so they are

neither terminating nor recurring decimals. neither terminating nor recurring decimals.

ﬃﬃ

77

ﬃ

p

_{is read as ‘the sq}_{is read as ‘the sq}_{uare root of 7’ or simply ‘}_{uare root of 7’ or simply ‘}_{root 7’.}_{root 7’.}

Rational numbers

Rational numbers such as fractions, terminating or recurring decimals, and percentages, can be such as fractions, terminating or recurring decimals, and percentages, can be expressed in the form

expressed in the form a a b

b;; where where a a and and b b are integers (and are integers (and b b

6

6 ¼

¼

0). Surds are0). Surds are irrational numbers irrational numbers

because they cannot be expressed in this form. because they cannot be expressed in this form.

Worksheet Worksheet StartUp assignment 13 StartUp assignment 13 MAT10NAWK10091 MAT10NAWK10091 Stage 5.3 Stage 5.3

(4)

Recurring decimals

Rational numbers

Rational numberscan be expressed in the formcan be expressed in the form

b

b Irrational numbersIrrational numberscannot cannot be expressed in the form be expressed in the form bb

2 2 3 3= 0.666 ...= 0.666 ... 5 5 6 6= 0.833 ...= 0.833 ... = 0.3636 ... = 0.3636 ... 4 4 11 11 –3 –3 1 1 = 26, = 26, = –3= –3 Integers Integers 26 26 1 1 4 4 1 1= 4,= 4,

Non-surds whose decimal value also have no

pattern and are non-recurring, for example,

π

π = 3.14159…, cos 38° = 0.7880..., = 3.14159…, cos 38° = 0.7880...,

0.0097542…

T

Transcendental ranscendental numbersnumbers

√ √ √√ √√ Surds Surds 5, 5, 2,2, , 8, 8 66 3 3 √ √1111 – – Terminating decimals Terminating decimals 0.5, 0.5, 7 7 = = 7.125,7.125, 16% = 0.16, 1.32 16% = 0.16, 1.32 1 1 8 8

E

x

a

m

p

l

e

1

Select the surds from this list of square roots:

p

ffiffiffiffiffi

56 56

ffiffiffiffiffiffiffiffi

p

135135

p

289289

ffiffiffiffiffiffiffiffi

p

9999

ffiffiffiffiffi

p

8181

Solution

ffiffiffiffiffi

56 56

p

¼

77::48334833. . .. . .

p

ffiffiffiffiffiffiffi

135135

ﬃ

¼

1111::61896189. . .. . .

p

ffiffiffiffiffiffiffi

289289

ﬃ

¼

1717

ffiffiffiffiffi

9999

p

¼

99::94989498. . .. . .

p

ffiffiffiffiffi

8181

¼

99

So the surds are

p

ffiffiffiffiffi

56 56,,

p

ffiffiffiffiffiffiffi

135135

ﬃ

andand

p

ffiffiffiffiffi

9999..

E

x

a

m

p

l

e

2

Is each number rational or irrational? Is each number rational or irrational? a a 37.5% 37.5% bb

_

p

44

ffiffiffiffiffiffiffi

216216

ﬃ

_c_c ₁₀₁₀ p p dd 00_:_:2266__ ee

p

ffiffiffiffiffi

4848

Solution

a a 3737:: 5 5%%

¼

3737:: 5 5 100 100

¼

3 3 8 8 [

[ 37.5% is a rational number. 37.5% is a rational number.

which is

which is in the in the form of form of a a fractionfraction a a

b

_

p

44

ffiffiffiffiffiffiffi

256256

ﬃ

_¼

_

44

[



p

44

ffiffiffiffiffiffiffi

256256

ﬃ

is is a a rational rational number.number.

which can

which can be written be written asas



44 1 1

c

c 1010pp

¼

31.415 926 5431.415 926 54 . . . . .. [

[ 10 10pp is an irrational number. is an irrational number.

The digits run endlessly without repeating. The digits run endlessly without repeating. d

d 00::2266__

¼

00::2666626666. . .. . .

¼

₁₅₁₅44

[

[ 0 0_:_:226 is a rational number.6 is a rational number.__

which is

which is a a recurring decimalrecurring decimal which is

which is a a fractionfraction

e

p

ffiffiffiffiffi

4848

_¼

66::9282032392820323. . .. . .

[

p

ffiffiffiffiffi

4848is is an an irrational irrational number.number.

The digits run endlessly without repeating. The digits run endlessly without repeating.

(5)

Square roots

The symbol

p

ﬃ

stands for the positive square root of a number. For examplstands for the positive square root of a number. For example,e,

p

ﬃﬃ

44

ﬃ

_¼

2 (not2 (not

_

2).2). Fu Furtrthehermrmorore,e,ititisisnonottpopossssibibleletotofinfinddththeesqsquauarereroroototofofaanenegagatitivevenunumbmberer..ItItisisononlylypoposssisiblbleetotofinfindd th theesqsquauarreeroroototofofaapoposisititivevenunumbmbererororzezeroro,,bebecacaususeeththeesqsquauarereofofananyyrerealalnunumbmbererisispoposisititiveveororzezeroro..

Summary

For

For x x > > 0, 0,

p

x x

ﬃﬃ

ﬃ

is the positiis the positive square roove square root of t of x x..

For

For x x

_¼

0,0,

p

x x

ﬃﬃ

ﬃ

is 0. is 0. For

For x x < < 0, 0,

p

x x

ﬃﬃ

ﬃ

is undeﬁned. is undeﬁned.

Surds on a number line

The rational and irrational numbers together make up the

The rational and irrational numbers together make up the real numbers real numbers. Any real number can be. Any real number can be represented by a point on the number line.

represented by a point on the number line.

– –₃₃ ––₂₂ ––₁₁ ₀₀ ₁₁ ₂₂ ₃₃ ₄₄ π π – – 33 10 10 –– 5 5__33 __ 120%120% 55 2 2 3 3



p

33

ffiffiffiffiffi

1010





22::15441544:::::: irrational irrational (surd)(surd)



₅₅33

¼



00::66 rational rational (fraction)(fraction)

2 2 3

3



00::66666666:::::: rational rational (fraction)(fraction)

120%

_¼

1.21.2 rational rational (percentage)(percentage)

ﬃﬃ

5 5

ﬃ

p



22::23602360:::::: irrational irrational (surd)(surd)

p



3.1415…3.1415… irrational irrational (pi)(pi)

E

x

a

m

p

l

e

3

Use a pair of compasses and Pythagoras’ theorem to estimate the value of

p

ﬃﬃ

22

ﬃ

on a numbon a number liner line.e.

Solution

Step 1 Step 1

Using a scale of 1 unit to 2 cm, Using a scale of 1 unit to 2 cm, draw a number line as shown.

draw a number line as shown. 0 0 1 1 22 33

Step 2 Step 2

Construct a right-angled triangle on Construct a right-angled triangle on the number line with base length and the number line with base length and height 1 unit as shown. By Pythagoras’ height 1 unit as shown. By Pythagoras’ theorem, show that

theorem, show that XZ XZ

_¼

p

ﬃﬃ

22

ﬃ

units. units.

0 0 11 1 1 2 2 33 1 1 X X Z Z 2 2 Step 3 Step 3

With 0 as the centre, use compasses with With 0 as the centre, use compasses with radius

radius XZ XZ



p

ﬃﬃ

22

ﬃ



to draw an arc to meet the to draw an arc to meet the number

number line at line at A A as shown. The point as shown. The point A A

represents the value of

p

ﬃﬃ

22

ﬃ

and and should should bebe approximately 1.4142… approximately 1.4142… 0 1 2 3 0 11 1 2 3 1 1 X X A A Z Z 2 2 Stage 5.3 Stage 5.3 Your calculato

Your calculator will r will tell you thattell you that

there is a

there is a mathemmathematical error if atical error if

you enter, for example,

p

ffiffiffiffiffiffi

_

5 5

ﬃ

::

Worksheet

Surds on the number

line

MAT10NAWK10092 MAT10NAWK10092

(6)

Exercise 1-01

Surds

and

irrational

numbers

1

1 Which one of thWhich one of the following is e following is a surd? Seleca surd? Select the correct at the correct answernswer A A,, B B,, C C or or D D.. A

A

p

ffiffiffiffiffi

6464 BB

p

ffiffiffiffiffiffiffi

100100

ﬃ

CC

p

ffiffiffiffiffiffiffi

250250

ﬃ

DD

p

ffiffiffiffiffiffiffi

400400

ﬃ

2

2 Which one of thWhich one of the following is e following is NOT a surd? NOT a surd? Select the corrSelect the correct answerect answer A A,, B B,, C C or or D D.. A

A

p

ffiffiffiffiffi

8484 BB

p

ffiffiffiffiffiffiffi

196196

ﬃ

CC

p

ffiffiffiffiffi

2727 DD

p

ffiffiffiffiffiffiffi

160160

ﬃ

3

3 Select the sSelect the surds from urds from the following the following list of list of square roots.square roots.

ffiffiffiffiffi

3232

p

₁₂₅

_{ffiffiffiffiffiffiffi}

₁₂₅

_ﬃ

p

_{ffiffiffiffiffiffiffiffi}

₆₂₅₆₂₅

p

_{ffiffiffiffiffiffiffiffi}

₄₀₀₄₀₀

p

₄₄

_{ffiffiffiffiffiffiffi}

_:_:₉₉

ffiffiffiffiffi

52 52

p

_{ffiffiffiffiffiffiffiffi}

₁₆₉₁₆₉

p

_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffi}

₀₀_:_:₀₀₀₉₀₀₀₉

p

₅₆₂₅₅₆₂₅

_{ffiffiffiffiffiffiffiffiffiffi}

p

_{ffiffiffiffiffiffiffiffi}

₂₈₈₂₈₈ 4

4 Is each nIs each number ratiumber rational (R) or onal (R) or irrational irrational (I)?(I)? a a 55:: _ _66 bb

p

ﬃﬃ

88

ﬃ

cc

p

ﬃﬃ

44

ﬃ

dd 3311 7 7 ee

ffiffiffiffiffi

2727 3 3

p

_f_f ₁₁_:_:₃₃₅₅__ g g

p

33

ffiffiffiffiffiffiffiffi

_

6464

ﬃ

hh 272711 2 2%% ii

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5 533101033

ﬃ

p

_j_j 33 11 11 kk

ffiffiffiffiffi

50 50

p

3 3 ll

p

ffiffiffiffiffiffi

p

ﬃﬃ

44

ﬃ

5

5 Arrange each Arrange each set of nuset of numbers in mbers in descending descending order.order. a a 1144 7 7;;

p

ﬃﬃ

22

ﬃ

;; p p 2 2 bb

ffiffiffiffiffi

2020 3 3

p

_;_;₂₂_:_:₆ _₆ __;_;₂₂77 9 9 6

6 Express each real numExpress each real number correct to one deciber correct to one decimal place and gramal place and graph them on a number linph them on a number line.e. a a

_

1144 5 5 bb 74%74% cc 4 4 11 11 dd



p

ffiffiffiffiffi

1212 e e

_

p

33

ffiffiffiffiffi

1515 f f 22 5 5 9 9 gg p p 2 2 hh 187% 187% 7

7 Use Use the mthe method ethod from from ExampleExample 3 3 to estimate the value of to estimate the value of

p

ﬃﬃ

22

ﬃ

on a on a number lnumber line.ine. 8

8 aa Use the mUse the method from ethod from ExampleExample 3 3 to estimate the value of to estimate the value of

p

ﬃﬃ

5 5

ﬃ

on a nuon a number mber line bline by y constructing a right-angled triangle with base length 2 units and height 1 unit. constructing a right-angled triangle with base length 2 units and height 1 unit. b

b Use a similar method to estimate the following surUse a similar method to estimate the following surds on a number line.ds on a number line. i

i

p

ffiffiffiffiffi

1010 iiii

p

ffiffiffiffiffi

1717

Investigation: Proof that

p

ﬃﬃ

₂

ﬃ

_{is irrational}

A method of proof sometimes used in mathematics is to assume the opposite of what is A method of proof sometimes used in mathematics is to assume the opposite of what is being proved, and show that it is false. This is called a

being proved, and show that it is false. This is called a proof by contradiction proof by contradiction.. We will use this method to prove that

We will use this method to prove that

p

ﬃﬃ

22 is irrational. is irrational.

ﬃ

Firstly, assume that

p

ﬃﬃ

22

ﬃ

isis rational rational . This means we assume that. This means we assume that

p

ﬃﬃ

22

ﬃ

can be can be written in written in thethe form

form a a b

b;; where where b b

¼

6

0, and0, and a a and and b b are integers with no common factor. are integers with no common factor.

ﬃﬃ

₂₂

ﬃ

p

¼

aa_b_b 2 2

_¼

aa 2 2 b

b22 Squaring Squaring both both sidessides

a

a22

_¼

22bb22

2

2bb22 is an even number because it is divisible by 2.is an even number because it is divisible by 2.

[

[ a a22 is even.is even.

See

See _{Example 1}_{Example 1}

See

(7)

1-02

Simplifying surds

Summary

For

For x x > > 0 (positive): 0 (positive):

ﬃﬃ

x x

ﬃ

p





22

_¼

p

ﬃﬃ

x x

ﬃ

33

p

ﬃﬃ

x x

ﬃ

_¼

x x

ffiffiffiffi

_x_x

ﬃ

22

p

¼

x x

E

x

a

m

p

l

e

4

Simplify each expression. Simplify each expression. a a



p

ffiffiffiffiffi

1212



22 bb



44

p

ﬃﬃ

77

ﬃ



22 cc



_

5 5

p

ﬃﬃ

22

ﬃ



22

Solution

a a



p

ffiffiffiffiffi

1212



22

_¼

1212 b b



44

p

ﬃﬃ

77

ﬃ



22

_¼

44

p

ﬃﬃ

77

ﬃ

3344

p

ﬃﬃ

77

ﬃ

44

p

ﬃﬃ

77

ﬃ

means 4 means 433

p

ﬃﬃ

77

ﬃ

¼

442233



p

ﬃﬃ

77

ﬃ



2 2

¼

16163377

¼

112112 c c



_

5 5

p

ﬃﬃ

22

ﬃ



22

_¼

_ð

5 5

_Þ

2233



p

ﬃﬃ

22

ﬃ



2 2

¼

25253322

¼

50 50 If

If a a22 is even, thenis even, then a a is also even because any odd number squared gives another odd is also even because any odd number squared gives another odd number.

number. If

If a a is even, then it is divisible by 2 and can be expressed in the form 2 is even, then it is divisible by 2 and can be expressed in the form 2mm, where, where m m is an is an integer. integer. ) ) aa22

¼

ð

22mm

Þ

22

¼

22bb22 4 4mm22

_¼

22bb22 2 2mm22

_¼

bb22 b b22

_¼

22mm22 [ [ b b22 is evenis even [

[ b b is even is even [

[ a a and and b b are both even. are both even.

This contradicts the assumption that

This contradicts the assumption that a a and and b b have no common factor. Therefore, the have no common factor. Therefore, the assumption that

assumption that

p

ﬃﬃ

22

ﬃ

is ratis rational is ional is false.false.

[

p

ﬃﬃ

22

ﬃ

must must be irbe irrational.rational.

1

1 Use the method of proof just described to show that these surds are irrationaUse the method of proof just described to show that these surds are irrational.l. a

a

p

ﬃﬃ

33

ﬃ

bb

p

ﬃﬃ

5 5

ﬃ

2

2 Compare your proofs wiCompare your proofs with those of other students.th those of other students. Stage 5.3 Stage 5.3 Puzzle sheet Puzzle sheet Simplifying surds Simplifying surds MAT10NAPS10093 MAT10NAPS10093 Technology worksheet Technology worksheet Excel worksheet: Excel worksheet:

Simplifying surds quiz

MAT10NACT00019 MAT10NACT00019 Technology worksheet Technology worksheet Excel spreadsheet: Excel spreadsheet: Simplifying surds Simplifying surds MAT10NACT00049 MAT10NACT00049

(8)

The square root of a product

For

For x x > > 0 and 0 and y y > > 0: 0:

ffiffiffiffi

xy xy

ﬃ

p

_¼

p

_ﬃﬃ

_x_x

_ﬃ

₃₃

p

_ﬃﬃ

_y_y

_ﬃ

A surd

p

ﬃﬃ

nn

ﬃ

can be can be simpliﬁed simpliﬁed if if n n can be divided into two factors, where one of them is a square can be divided into two factors, where one of them is a square number such as 4, 9, 16, 25, 36, 49, …

number such as 4, 9, 16, 25, 36, 49, …

E

x

a

m

p

l

e

5

Simplify each surd. Simplify each surd. a a

p

ﬃﬃ

88

ﬃ

bb

p

ffiffiffiffiffiffiffi

108108

ﬃ

cc 44

p

ffiffiffiffiffi

4545 dd

p

ffiffiffiffiffiffiffi

288288

ﬃ

3 3

Solution

a a

p

ﬃﬃ

₈₈

ﬃ

_¼

p

ﬃﬃ

₄₄

ﬃ

₃₃

p

ﬃﬃ

₂₂

ﬃ

¼

2233

p

ﬃﬃ

22

ﬃ

¼

22

p

ﬃﬃ

22

ﬃ

4 is a square number. 4 is a square number. b b MeMeththod od 11

ffiffiffiffiffiffiffi

₁₀₈₁₀₈

ﬃ

p

¼

p

ffiffiffiffiffi

363633

p

ﬃﬃ

33

ﬃ

¼

6633

p

ﬃﬃ

33

ﬃ

¼

66

p

ﬃﬃ

33

ﬃ

Method 2 Method 2

ffiffiffiffiffiffiffi

₁₀₈₁₀₈

ﬃ

p

¼

p

ﬃﬃ

44

ﬃ

33

p

ffiffiffiffiffi

2727

¼

2233

p

ﬃﬃ

99

ﬃ

33

p

ﬃﬃ

33

ﬃ

¼

22333333

p

ﬃﬃ

33

ﬃ

¼

66

p

ﬃﬃ

33

ﬃ

Method 2 involves simplifying surds

Method 2 involves simplifying surds twice twice ( (

p

ffiffiffiffiffiffiffi

108108

ﬃ

andand

p

ffiffiffiffiffi

2727). M). Method 1 ethod 1 shows thshows thatat when

when simplifying surds, you simplifying surds, you should look should look for the for the highest square highest square factor possible.factor possible. c c 44

p

ffiffiffiffiffi

4545

_¼

4433

p

ﬃﬃ

99

ﬃ

33

p

ﬃﬃ

5 5

ﬃ

¼

44333333

p

ﬃﬃ

5 5

ﬃ

¼

1212

p

ﬃﬃ

5 5

ﬃ

d d

p

ffiffiffiffiffiffiffi

288288

ﬃ

3 3

¼

ffiffiffiffiffiffiffi

144144

ﬃ

p

₃₃

p

_ﬃﬃ

₂₂

_ﬃ

3 3

¼

1212

ﬃﬃ

22

ﬃ

p

3 3

¼

1212 4 4

_ﬃﬃ

_ﬃ

2 2

p

3 311

¼

44

p

ﬃﬃ

22

ﬃ

Video tutorial Video tutorial Simplifying surds Simplifying surds MAT10NAVT10002 MAT10NAVT10002

(9)

Stage 5.3

_{Exercise 1-02}

_Simplifyin

_g

_surds

1

1 Simplify Simplify each each expression.expression. a

a



p

ﬃﬃ

22

ﬃ



22 bb



p

ﬃﬃ

5 5

ﬃ



22 cc



33

p

ﬃﬃ

33

ﬃ



22 dd



55

p

ffiffiffiffiffi

1010



22

e



p

ffiffiffiffiffiffiffiffi

00::0909

ﬃ



22 f f



22

p

ﬃﬃ

77

ﬃ



22 gg



33

p

ﬃﬃ

5 5

ﬃ



22 hh



5 5

p

ﬃﬃ

22

ﬃ



22

2

2 Simplify Simplify each each surd.surd. a a

p

ffiffiffiffiffi

50 50 bb

p

ffiffiffiffiffi

1212 cc

p

ffiffiffiffiffi

2828 dd

p

ffiffiffiffiffiffiffi

150150

ﬃ

ee

p

ffiffiffiffiffiffiffi

700700

ﬃ

f f

p

ffiffiffiffiffi

4545 gg

p

ffiffiffiffiffi

4848 hh

p

ffiffiffiffiffiffiffi

200200

ﬃ

ii

p

ffiffiffiffiffi

9696 jj

p

ffiffiffiffiffi

6363 k k

p

ffiffiffiffiffiffiffi

288288

ﬃ

ll

p

ffiffiffiffiffiffiffi

108108

ﬃ

mm

p

ffiffiffiffiffi

7575 nn

p

ffiffiffiffiffiffiffi

147147

ﬃ

oo

p

ffiffiffiffiffi

3232 p p

p

ffiffiffiffiffiffiffi

242242

ﬃ

qq

p

ffiffiffiffiffiffiffi

162162

ﬃ

rr

p

ffiffiffiffiffiffiffi

245245

ﬃ

ss

p

ffiffiffiffiffiffiffi

125125

ﬃ

t t

p

ffiffiffiffiffiffiffi

512 512

ﬃ

3

3 Simplify Simplify each each expression.expression. a a 33

p

ffiffiffiffiffi

2020 bb 44

p

ffiffiffiffiffi

3232 cc 88

p

ffiffiffiffiffi

7272 dd

p

ffiffiffiffiffi

4040 2 2 ee

ffiffiffiffiffiffiffi

243243

ﬃ

p

9 9 f f

p

ffiffiffiffiffi

2828 6 6 gg 33

p

ffiffiffiffiffi

2424 hh 99

p

ffiffiffiffiffi

6868 ii

ffiffiffiffiffiffiffiffiffi

31253125

ﬃ

p

10 10 jj 1 1 2 2

ffiffiffiffiffi

7272

p

k k 33 4 4

ffiffiffiffiffi

4848

p

_l_l ₁₀₁₀

p

_{ffiffiffiffiffiffiffi}

₁₆₀₁₆₀

_ﬃ

_m_m ₃₃

p

_{ffiffiffiffiffi}

₇₅₇₅ _n_n ₇₇

p

_{ffiffiffiffiffi}

₆₈₆₈ _o_o

p

ffiffiffiffiffi

52 52 6 6 4

4 Which one Which one of the foof the following is llowing is equivalent to equivalent to 44

p

ffiffiffiffiffi

50 50? ? SelectSelect A A,, B B,, C C or or D D.. A

A 88

p

ﬃﬃ

5 5

ﬃ

BB 2020

p

ﬃﬃ

22

ﬃ

CC 88

p

ﬃﬃ

22

ﬃ

DD 2020

p

ﬃﬃ

5 5

ﬃ

5

5 Which onWhich one of e of the followinthe following is g is equivalent tequivalent too

p

ffiffiffiffiffiffiffi

250250

ﬃ

10

10 ? ? SelectSelect A A,, B B,, C C or or D D.. A A

p

ﬃﬃ

5 5

ﬃ

10 10 BB

ffiffiffiffiffi

1010

p

2 2 CC 22

p

ffiffiffiffiffi

1010 DD 55

p

ffiffiffiffiffi

1010 6

6 Decide whethDecide whether each ser each statement is tatement is true (T) true (T) or false or false (F).(F). a

a 33

p

ﬃﬃ

5 5

ﬃ

_¼

p

ffiffiffiffiffi

1515 bb

p

ffiffiffiffiffi

1818

_¼

99 cc



p

ffiffiffiffiffiffi

99::44

ﬃ



22

¼

99::44 dd

p

ffiffiffiffiffi

7575

¼

5 5

p

ﬃﬃ

33

ﬃ

e

p

ﬃﬃ

33

ﬃ

_

11::77 f f The eThe exact xact value value of of

p

ffiffiffiffiffi

1010 is is 3.162 3.162 277 277 88

Just for the record

Unreal

numbers

are

imaginary!

There exist numbers that are neither

There exist numbers that are neither rational rational nor nor irrational irrational, so they are also not, so they are also not real numbers real numbers.. For example,

For example,

p

ffiffiffiffiffiffi

_

22

ﬃ

is not a is not a real numbereal number, because r, because there is there is no real no real number whicnumber which, if squh, if squared,ared, equals

equals

_

2. Numbers such as2. Numbers such as

p

ffiffiffiffiffiffi

_

22

ﬃ

;;

p

ffiffiffiffiffiffiffiffi



1010

ﬃ

andand

p

44

ffiffiffiffiffiffiffiffi



1717

ﬃ

are are calledcalled unreal unreal or or imaginary numbers imaginary numbers

and cannot be graphed on a number line (that is, their values cannot be ordered). and cannot be graphed on a number line (that is, their values cannot be ordered). Imaginary numbers were ﬁrst noticed by Hero of Alexandria in the 1st century

Imaginary numbers were ﬁrst noticed by Hero of Alexandria in the 1st century CECE. In 1545,. In 1545,

the Italian mathematician Girolamo Cardano wrote about them, but believed negative the Italian mathematician Girolamo Cardano wrote about them, but believed negative numbers did not have a square root. Imaginary numbers were largely ignored until the 18th numbers did not have a square root. Imaginary numbers were largely ignored until the 18th century when they were studied by Leonhard Euler and the Carl Friedrich Gauss.

century when they were studied by Leonhard Euler and the Carl Friedrich Gauss.

ffiffiffiffiffiffi

_

11

ﬃ

p

_is_is_deﬁned_deﬁned_to_to_be_be_the_the_imaginary_imaginary_number_number i_i_,_,_so_so

_{ffiffiffiffiffiffi}

_ﬃ



11

p

¼

ii.. ) )

p

ffiffiffiffiffiffiffiffi



3636

ﬃ

¼

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

363633

ð



11

Þ

ﬃ

¼

p

ffiffiffiffiffi

363633

p

ffiffiffiffiffiffi



11

ﬃ

¼

66ii_:_:

Imaginary numbers are useful for solving physics and engineering problems involving heat Imaginary numbers are useful for solving physics and engineering problems involving heat conduction, elasticity, hydrodynamics and the ﬂow of electric current.

conduction, elasticity, hydrodynamics and the ﬂow of electric current. Simplify each imaginary number.

Simplify each imaginary number. a

a

p

ffiffiffiffiffiffiffiffiffiffiffi

_

100100 bb

p

ffiffiffiffiffiffiffiffi

_

2525

ﬃ

cc

p

44

ffiffiffiffiffiffiffiffi

_

1616

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dd

p

66

ffiffiffiffiffiffiffiffi

_

6464

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See

(10)

1-03

Adding and subtracting surds

Just as

Just as you can you can only add only add or or subtract ‘like subtract ‘like terms’ in terms’ in algebra, you algebra, you can only can only add or add or subtract ‘likesubtract ‘like surds’. You may ﬁrst need to express all the surds in their simplest forms.

surds’. You may ﬁrst need to express all the surds in their simplest forms.

E

x

a

m

p

l

e

6

Simplify each expression. Simplify each expression. a a 55

p

ffiffiffiffiffi

1111

_þ

77

p

ffiffiffiffiffi

1111 bb 88

p

ﬃﬃ

5 5

ﬃ

_

33

p

ﬃﬃ

5 5

ﬃ

cc 33

p

ﬃﬃ

66

ﬃ

_

44

p

ﬃﬃ

22

ﬃ

_þ

5 5

p

ﬃﬃ

66

ﬃ

d d

p

ffiffiffiffiffi

8080

_þ

p

ffiffiffiffiffi

2020 ee

p

ﬃﬃ

88

ﬃ

_

p

ffiffiffiffiffi

2727

_þ

p

ffiffiffiffiffi

1818 f f 55

p

ffiffiffiffiffi

2020

_

33

p

ffiffiffiffiffiffiffi

125125

ﬃ

Solution

a a 55

p

ffiffiffiffiffi

1111

_þ

77

p

ffiffiffiffiffi

1111

_¼

1212

p

ffiffiffiffiffi

1111 bb 88

p

ﬃﬃ

5 5

ﬃ

_

33

p

ﬃﬃ

5 5

ﬃ

_¼

5 5

p

ﬃﬃ

5 5

ﬃ

c c 33

p

ﬃﬃ

66

ﬃ

_

44

p

ﬃﬃ

22

ﬃ

_þ

5 5

p

ﬃﬃ

66

ﬃ

_¼

88

p

ﬃﬃ

66

ﬃ

_

44

p

ﬃﬃ

22

ﬃ

d d

p

ffiffiffiffiffi

₈₀₈₀

_þ

p

ffiffiffiffiffi

₂₀₂₀

_¼

p

ffiffiffiffiffi

₁₆₁₆

p

₅₅

ﬃﬃ

ﬃ

_þ

p

ﬃﬃ

₄₄

ﬃ

p

₅₅

ﬃﬃ

ﬃ

¼

44

p

ﬃﬃ

5 5

ﬃ

_þ

22

p

ﬃﬃ

5 5

ﬃ

¼

66

p

ﬃﬃ

5 5

ﬃ

Simplifying each surd. Simplifying each surd.

e e

p

ﬃﬃ

₈₈

ﬃ

_

p

ffiffiffiffiffi

₂₇₂₇

_þ

p

ffiffiffiffiffi

₁₈₁₈

_¼

p

ﬃﬃ

₄₄

ﬃ

p

₂₂

ﬃﬃ

ﬃ

_

p

ﬃﬃ

₉₉

ﬃ

p

₃₃

ﬃﬃ

ﬃ

_þ

p

ﬃﬃ

₉₉

ﬃ

p

ﬃﬃ

₂₂

ﬃ

¼

22

p

ﬃﬃ

22

ﬃ

_

33

p

ﬃﬃ

33

ﬃ

_þ

33

p

ﬃﬃ

22

ﬃ

¼

5 5

p

ﬃﬃ

22

ﬃ

_

33

p

ﬃﬃ

33

ﬃ

f f 55

p

ffiffiffiffiffi

2020

_

33

p

ffiffiffiffiffiffiffi

125125

ﬃ

_¼

5 5

p

ﬃﬃ

44

ﬃ

p

5 5

ﬃﬃ

ﬃ

_

33

p

ffiffiffiffiffi

2525

p

5 5

ﬃﬃ

ﬃ

¼

5 53322

p

ﬃﬃ

5 5

ﬃ

_

3333 5 5

p

ﬃﬃ

5 5

ﬃ

¼

1010

p

ﬃﬃ

5 5

ﬃ

_

1515

p

ﬃﬃ

5 5

ﬃ

¼



5 5

p

ﬃﬃ

5 5

ﬃ

Exercise 1-03

Adding

and

subtracting

surds

1

p

ﬃﬃ

33

ﬃ

_þ

22

p

ﬃﬃ

33

ﬃ

bb 1111

p

ﬃﬃ

22

ﬃ

_

88

p

ﬃﬃ

22

ﬃ

cc 55

p

ﬃﬃ

66

ﬃ

_

p

ﬃﬃ

66

ﬃ

d d

p

ﬃﬃ

5 5

ﬃ

_þ

33

p

ﬃﬃ

5 5

ﬃ

ee 55

p

ffiffiffiffiffi

1717

_

5 5

p

ffiffiffiffiffi

1717 f f 33

p

ffiffiffiffiffi

1010

_

22

p

ffiffiffiffiffi

1010 g g 44

p

ffiffiffiffiffi

1515

_

33

p

ffiffiffiffiffi

1515

_þ

77

p

ffiffiffiffiffi

1515 hh 55

p

ﬃﬃ

66

ﬃ

_

22

p

ﬃﬃ

66

ﬃ

_

44

p

ﬃﬃ

66

ﬃ

ii 33

p

ﬃﬃ

33

ﬃ

_þ

44

p

ﬃﬃ

33

ﬃ

_

5 5

p

ﬃﬃ

33

ﬃ

j j 44

p

ﬃﬃ

5 5

ﬃ

_þ

77

p

ﬃﬃ

5 5

ﬃ

_

p

ﬃﬃ

5 5

ﬃ

kk 88

p

ffiffiffiffiffi

1010

_

5 5

p

ffiffiffiffiffi

1010

_þ

33

p

ffiffiffiffiffi

1010 ll 1010

p

ﬃﬃ

33

ﬃ

_

33

p

ﬃﬃ

33

ﬃ

_

1212

p

ﬃﬃ

33

ﬃ

2

p

ﬃﬃ

ﬃ

33

_

99

_þ

22

p

ﬃﬃ

33

ﬃ

bb 1111

p

ffiffiffiffiffi

1010

_

77

p

ﬃﬃ

22

ﬃ

_

44

p

ffiffiffiffiffi

1010 c c

_

44

p

ﬃﬃ

33

ﬃ

_þ

5 5

p

ﬃﬃ

22

ﬃ

_

5 5

p

ﬃﬃ

33

ﬃ

dd 33

p

ffiffiffiffiffi

1515

_þ

33

p

ﬃﬃ

22

ﬃ

_þ

44

p

ffiffiffiffiffi

1515

_þ

5 5

p

ﬃﬃ

22

ﬃ

e e

p

ﬃﬃ

77

ﬃ

_

33

p

ﬃﬃ

5 5

ﬃ

_

44

p

ﬃﬃ

77

ﬃ

_þ

p

ﬃﬃ

5 5

ﬃ

f f 44

p

ﬃﬃ

66

ﬃ

_

33

p

ﬃﬃ

33

ﬃ

_

22

p

ﬃﬃ

66

ﬃ

_

5 5

p

ﬃﬃ

33

ﬃ

g g 1010

p

ffiffiffiffiffi

1111

_

5 5

p

ﬃﬃ

33

ﬃ

_þ

33

p

ffiffiffiffiffi

1111

_þ

44

p

ﬃﬃ

33

ﬃ

hh

p

ffiffiffiffiffi

1313

_þ

88

p

ﬃﬃ

77

ﬃ

_

77

p

ffiffiffiffiffi

1313

_þ

33

p

ﬃﬃ

77

ﬃ

i i 22

p

ﬃﬃ

5 5

ﬃ

_

33

p

ﬃﬃ

77

ﬃ

_

22

p

ﬃﬃ

5 5

ﬃ

_

33

p

ﬃﬃ

77

ﬃ

jj 44

p

ffiffiffiffiffi

1010

_

33

p

ﬃﬃ

5 5

ﬃ

_

44

p

ffiffiffiffiffi

1010 3

3 For each For each expression, expression, select the select the correct simcorrect simpliﬁed answepliﬁed answerr A A,, B B,, C C or or D D.. a a