9th mathematics exercise question with answer

# 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  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

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.

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

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 – 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 – 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