Integrated Mathematics

Surds

Irrational Number•Number that cannot be expressed as a fraction of two integers

2 6

2

1

27

1

2

3 27

1

Think!Which of the following is NOT a irrational number?

26

36

1

25

1

55

9

1)3

1( 2 x x

x

Irrational number

Irrational number

Rules of Indices and Surds 1mma a

mmm baba111

)(

m m ma b ab

m

mm

a a

bb

mmm abba111

)(

a a a 1 1

2 2a a a

a b a b a b a b

Simplify

24327

9 3 81 3

3933

38139

312

m m mab a b

Simplify

28112175

25 7 16 7 4 7

727475

74716725

77

Simplify

271234

933434

3)24(3

3

38

333234

Simplify

22332

30 12 6

)2)(9(632234

1861212

222 2)( bababa 23 32 2332 32 23

Simplify

223332

33325

3215

)132(5

Rationalization of the Denominator

•Process of removing a surd from the denominator

•Example:2

1

2

2

2

1

22

2

12

1

2

2

2

2

•Example:23

2

23

23

23

2

2323

232

123

2

22

23

232

23

23

Multiple together!

1

432

43

232

432

Objective is to remove surds from

denominator

Conjugate Surds

Note!

22))(( bababa

23

23

23

2

23

2

Re-look into the question:

2323

232

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

22

23

232

3241

432

27

27

27

2

27

2

Rationalize the denominators:

2727

272

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

22

27

272

47

272

3

272

13

13

13

33

13

33

Rationalize the denominators:

1313

1333

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

22

2

13

33333

13

3323

32

32

23

21019

932

3242616

324

324122

324

324

324

122

324

324

324

122

22

324

122

Rationalize the denominators:

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

3223

3223

3223

3223

3223

3223

Rationalize the denominators:

32233223

32233223

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

22

2

3223

3223

22

22

3223

323223223

6256

61230

3429

3461229

mmm abba111

)(

3253

5237

Rationalize the denominators:

Multiply conjugate surds

Multiply denominator and

numerator

22))(( bababa

33

721525

1245

15430421521

32533253

32535237

3253

3253

3253

5237

73

1

73

1

Rationalize the denominators:

Determine the LCM

Multiply denominator and

numerator

73

1

73

1

73

73

7373

73

7373

73

73

73

7373

7373

7373

2

3

4

32

73

32

)154(2

53

)154(2

53

22 )53(

53

)35(

53

2222 )5(532)3(

53

)3(352)5(

53

Rationalize the denominators:

Expand the denominators

51523

53

31525

53

1528

53

1528

53

Find the LCM

)154(

)154(

)154(2

53

)154(

)154(

)154(2

53

)1516(2

)755415312()755415312(

)1(2

)7512(2 3512

1541542

524

23235

524

23

524

235

Rationalize the denominators:

Same denominator

Rationalization

2016

1042851632

524

228

)524)(524(

)524)(228(

10225484

1042851632

524

524