Surds – Full Chapter Surds – Class IX

1.3.1 Introduction – Surds

You have studied about the irrational numbers like √3, ∛7, ∜5 and so on. You have here square root, cube rot and forth root of numbers. There are also irrational numbers which cannot be written in such a form; for example √(3+∛2), π. In the Surds we study more about numbers of the form Surds is a natural number. ‘a’ is a rational number. these are special class of irrational numbers.

1.3.2 Rational Exponent of a Number – Surds

Recall the statement from unit 2: Real Numbers given a natural number n, for every positive real number a there exists a unique positive real number b such that bn  = a. Here b is called the n-th root of a and we write

Surds

We can now define rational power of a positive number. let a be a positive real number. Let r = p/q to be a rational number, where p is an integer and q is a natural number.

Surds

Let a and b be positive real numbers. Let r1 and r2 be two rational numbers. Then we have:

Surds

Example 1: Simplify 2^1/2/4^1/6

Solution:

41/6 = (22)1/6  = 22x(1/6) = 21/3

Hence,

2^1/2/4^1/6 = 2^1/2/2^1/3 = 2(1/2)-(1/3) = 21/6


Surds – Exercise 1.3.2

  1. Simplify the following using laws of indices:

(i) (16)-0.75 x (64)4/3

(16)-0.75 x (64)4/3                              [16 = 24 ]

= (24)-0.75 x (24)4/3                         [64 = 26 ]

= 24 x -3/4 x 26 x 4/3

= 2-3 x 28

= 25

= 32


(ii) (0.25)0.5 x (100)-1/2

(0.25)0.5 x (100)-1/2

= (0.25)1/2 x (1/100 )1/2

= (0.25)1/2 x (1/102 )1/2

= (0.5) x ( 1/10 )

= ( 5/10 ) x ( 1/10)

= 5/100

= 𝟏/𝟐𝟎


(iii) (6.25)0.5 x 102 x (100)-1/2 x (0.01)-1

= ( 625/100 )1/2 x 102 x (1/10)1/2 x ( 1/100 )-1

= ( 25/10 )1/2 x 102 x (1/10 )1/2 x (1/100)-1

= ( 252/10 )1/2 x 102 x (1/102 )1/2 x (100)1

= ( 25/10 ) x 100 x 1 10 x 100

= 2500


(iv) (3-1/2 x 2-1/3) ÷ (3-3/4 x 2-5/6)

= 3−1/2× 2−1/3 3−3/4× 2−5/6

= 3-1/2+3/4 x 2-1/3+5/6

= 31/4 x 23/6

= 31/4 x 21/2

= 31/4 x 21/4

=(3 × 22)1/4

= (3 × 4)1/4

= 𝟏𝟐1/4


  1. Find the value of the expression.

[ 31/3 {5-1/2 x 3-1/3 x (2252)1/3}1/2]6

Solution:

= [ 31/3 {5-1/2x-1/2 x 3-1/3x1/2 x (2252)2/3x1/2]6

= [ 31/3 {5-1/4 x 31/6 x (2252)-2/6]6

= [ 31/3 x 6 {51/4 x 6 x 31/6 x 6 x (2252)-2/6 x 6]

= [ 32 x 51/4 x 6 x 31/6 x 6 x 2252-2/6 x 6]

= [32 x 53/2 x 31 x 225-2]

= [32 x 52/3 x 31 x 15-4]

= [32 x 5-2/3 x 31 x 3-4 x 5-4]

= [32 + 1 – 4 x 53/2 – 4]

= 1/31 x 1/55/2

= 1/31  x 1/55/2

= 1/31  x 1/√55

= 1/3 x 1/√3125

= 1/3√3125


  1. Simplify:

[{(35/2 x 53/4) ÷ 2-5/4} ÷ {16 / (52 x 21/4 x 31/2)}]1/5

Solution:

= [{(35/2 × 53/4) ÷ 2−5/4} ÷ {16÷ (52× 21/4× 31/2)}] 1/5

= [{(35/2 × 53/4)/(2−54) ÷ 16/(52× 21/4× 31/2)}] 1/5

= [(35/2 × 53/4)/(2−54) ÷ (52× 21/4× 31/2)/24]1/5

= [(35/2 × 53/4 × 52/1 × 21/4 × 31/2)/(2−5/4 × 24 )]1/5

= [21/4 × 35/2 + 1/2 × 53/4 + 2/1/2−5/4 + 4/1]1/5

= 21/4 × 35/2 + 1/2 × 511/4/211/4]1/5

= [21/4 4/11 × 33 × 511/4 ] 1/5

= [210/4 × 1/5 × 33 × 1/5 × 511/4 + 1/5]

= 21/2 × 31/5 × 511/20


1.3.3 Surds and their properties – Surds:

Consider the following real numbers:

√17, 8 + ∛12 , 3/5 + √(7/11) , ∛(3 + √5))

They are all irrational numbers. Nevertheless, you see that they are all different types.

A surd is real number of the form  , where n is an integer larger than 1 and a is a rational number such that it is not an n-th power of any rational number. For example, 25/36 is the square of 5/6. Thus, √(5/6) is not a surd. On the other hand √(24/17) is a surd.

A simplest form of a surd:

If , where c does not contain any n-th power of a rational number, then it is the simplest form. Here b is the coefficient of the surd.

Pure surd:

A surd which in its simplest form has 1 as coefficient.

Mixed Surd:

A surd which has some rational number not equal to 1 as its coefficient written its simplest form.

Similar surds:

Two surds are called similar surds or like surds if when written in simplest form they have the same order and the same radicand. Otherwise they are called unlike surds.

Reduction of Surds of different orders to the same order:

Example: Reduce the surds √(24/98) and ∛16 to the surds of the same order

Solution:

Here the order are 2 and 3 respectively. Hence their LCM is 6. We write

Surds


Surds – Exercise 1.3.3

  1. Write the following surds in their simplest form:

(i) √𝟕𝟔

= √(19 × 4)

= √(19 ×24)

= 𝟐√𝟏𝟗

 

(ii) (𝟏𝟎𝟖)

= ∛(33 × 22)

= ∛22

= ∛𝟒

 

(iii) 𝟓𝟎𝟎𝟎

= 5000

= ∜(511 × 84)

= 𝟓𝟖

 

(iv) 𝟏𝟖𝟗/𝟐𝟓

= ∛[(33 × 7)/52]

= 3𝟕/𝟐𝟓

 

(v) 𝟒𝟎𝟎/𝟒𝟗

= ∜(24 × 52)/724

= 252/72

= 2𝟐𝟓/𝟒𝟗


  1. Classify the following in to like surds

(i) √𝟐𝟒 , √𝟏𝟐𝟖 , √𝟕𝟓 , √𝟕𝟐 , √𝟓𝟒 , √𝟐𝟒

= 35 , 27 , √(52×3) , √(32×23) , √(2 × 33) , √(23×3)

= 323 , 232 , 53 , 3×22 , 3(2×3), 2(2×3)

= 93, 82, 53, 62, 36, 26

= {82, 62}, {93, 53}, {26, 36,}

={ √𝟏𝟐𝟖, √𝟕𝟐}, { √𝟐𝟒𝟑, √𝟕𝟓}, { √𝟓𝟒, √𝟐𝟒,}


(ii) ∛𝟐𝟎𝟎𝟎, ∛𝟔𝟖𝟔, ∛𝟔𝟒𝟖, ∛𝟑𝟕𝟓, ∛𝟏𝟐𝟖, ∛𝟐𝟒

= ∛(2×52), ∛(73×2), ∛(63× 3), ∛(53×3), ∛(43×2), ∛(23× 3)

= 10∛2, 7∛2, 6∛3, 5∛3, 4∛2, 2∛3

= {4∛𝟐, 7∛𝟐, 10∛𝟐} & {2∛𝟑, 5∛𝟑, 6∛𝟑}

= { ∛𝟐𝟎𝟎𝟎, ∛𝟔𝟖𝟔 , ∛𝟏𝟐𝟖} & { ∛𝟔𝟒𝟖, ∛𝟑𝟕𝟓, ∛𝟐𝟒}


  1. Which of the following are pure surds?

(i) 296 = √(33×37) = 2√(37×2) = 2√𝟕𝟒 – not pure surds

(ii) 729 = 36 = 33 = 27 –  not pure surds

(iii) ∛ 211 Cannot be reduced further hence it is a pure surd. – Yes, a pure surd

(iv) 75 is also a pure surd. – Yes, a pure surds

(v) ∛ 296  = ∛(23×37) = 2 ∛𝟑𝟕 – no, not a pure surd

(vi) 296 = cannot be reduced further, hence it is a pure surd – Yes


  1. Write the following irrational numbers sure from

(i) √(𝟏𝟓√(𝟐𝟕))

= 27 = 33 = 33

= [15(27)1/2]1/2

= 152/4 x 271/4             [1/2 = 4/2]

= ∜(153 × 27)

= ∜(225 × 27)

= ∜𝟔𝟎𝟕𝟓

 

(ii) √(𝟒𝟎(𝟏𝟐))

= [40 × (12)1/2] ½

= (40)1/2 × (12)1/4

= ∜(402 × (12)1)

= ∜(402× 121)

= ∜(1600×12)

= ∜19200

 

(iii) √(𝟓(𝟒𝟖))

= [51/2 ×(48)1/2]½

= [51/2 × 481/4]

= ∜ (52× 48)

= ∜(25×48)

= ∜1200


5. Reduce the following to surds of the same order.

(i) , √𝟐and 𝟓1/5

The orders are 3, 2, and 5

LCM of 2, 3 and 5 is 30

∛2 = 21/3 = 210/30 = (𝟐𝟏𝟎) 1/3𝟎

2 = 21/2 = 215/30 = (𝟐𝟏𝟓 ) 1/3𝟎

51/5 = 51/5 = 56/30 = (𝟓𝟔) 1/3𝟎

(1024)1/3𝟎 , (32768) 1/3𝟎 , (15625) 1/3𝟎

They have the same order 30

 

(ii) √𝟓 , (√𝟏𝟓)1/4 , and (√𝟓𝟎)1/8

Order is 2, 4, 8

Their LCM is 8

5 = 51/2 = 54/8 = (54 )1/8 = (125)1/8

(√15)1/4 = (15)1/4 = (15)2/8 = ((15)2) 1/8 = (225)1/8

(50)1/8 is in its simplest form

(50)1/8  , 𝟐𝟐𝟓𝟖 , 𝟏𝟐𝟓𝟖

∴ Thus they all are of the same order 8.

 

(iii) √𝟐 , √𝟕1/3 , (√𝟏𝟏)1/4 and (√𝟏𝟔𝟕𝟏)𝟏/1𝟐

Order is 2, 3, 4, and 12

Their LCM is 12

2 = 21/2 = 26/12 = (26)1/12 = (64)1/12

71/3 = 74/12 = (74 )1/12 = (2401)1/12

(√11)1/4 = (11)3/12 = ((11)3)1/12 = (1331)1/12

167112 is in its simplest form

𝟔𝟒𝟏/1𝟐 , 𝟐𝟒𝟎𝟏𝟏/1𝟐 , 𝟏𝟑𝟑𝟏𝟏/1𝟐 , 𝟏𝟔𝟕𝟏𝟏/1𝟐

∴ Thus they all are of the same order 12


1.3.4 Comparing Surds and Some Irrational Numbers:

Example 7: Find which surd is larger

Surds


Surds – Exercise 1.3.4

  1. Find which is larger:

(i) 3𝟑 and 4𝟒

∛(3×32)                                                        4×424

= 27 x 3                                                   = 256 x 4

= 81                                                             = 1024

= ((81)4)1/12                                                  = ((1024)4)1/12

= (43046721)1/12                                         = (1073741824)1/12

4𝟒 is greater 3𝟑


  1. Compare the following and decide which is larger.

(i) (∜(30))1/7 and ∛(281/10)

(281/10) 1/3 = (281/10)1/3 = 281/30

(301/4) 1/7 = (301/4)1/7 = 301/28

LCM of 30 and 28 is 420

(281/10)1/3 = 281/30 = (281/14)1/420

(301/4) 1/7 = 301/28 = (3015)1/420

2814 =22 x 714 = 228 x 714

3015 = (5 x 6)15 = 515 x 615

Comparing the 2 numbers we conclude

3015 > 2814

(∜(30))1/7 > ∛(281/10)

 

(ii) √(∜8)  and ∛(∛9)

√(∜8) = (81/4)1/2 = 81/8 = 23/8          [8 = 23]

∛(∛9) = (91/3)1/3 = (32/3)1/3 = 32/9

LCM of 8 and 9 is 72

√(∜8)  = 23/8 = (23/8)9/9 = 227/72 = (227)1/72

∛(∛9) = 32/9 = (32/9)8/8 = 316/72 = (316)1/72

By comparing we find that (227) is larger than 316.

Hence √(∜8)   > ∛(∛9)


  1. Write the following in ascending order:

√𝟐 , 𝟑𝟑 , 𝟔1/6

21/2 , 31/3 , 61/6

LCM of 2, 3 and 6 is 6

21/2 = 23/6 = (23)1/6 = 81/6

31/3 = 32/6 = (33)1/6 = 91/6

61/6 = 63/6 = (6)1/6 = 61/6

Ascending order is 61/6  , 21/2, 31/3


  1. Write the following descending order:

√(𝟔) , (∜𝟏𝟐) , (∜𝟖)

(61/3)1/2 , (121/4)1/3 , (81/4)1/2

61/6, 121/12 , 81/8

LCM of 6, 12, 8 is 24

64/24, 122/24, 83/24

(64)1/24, (122)1/24, (83)1/24

(1296)1/24, (144)1/24, (512)1/24

Descending order is √(𝟔)  , √ (∜𝟖)  , ∛(∜𝟏𝟐)


Real Numbers

Square root