Mathematics

SURDS – EXERCISE 1.3.3 – Class 9

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