Mathematics

SURDS – EXERCISE 1.3.3 – Class 9

  1. Write the following surds in their simplest form:

(i) βˆšπŸ•πŸ”

= √(19 Γ— 4)

= √(19 Γ—24)

= πŸβˆšπŸπŸ—

 

(ii) βˆ› (πŸπŸŽπŸ–)

= βˆ›(33 Γ— 22)

= βˆ›22

= βˆ›πŸ’

 

(iii) βˆœπŸ“πŸŽπŸŽπŸŽ

= ∜5000

= ∜(511 Γ— 84)

= πŸ“βˆœπŸ–

 

(iv) βˆ›πŸπŸ–πŸ—/πŸπŸ“

= βˆ›[(33 Γ— 7)/52]

= 3βˆ›πŸ•/πŸπŸ“

 

(v) βˆœπŸ’πŸŽπŸŽ/πŸ’πŸ—

= ∜(24 Γ— 52)/724

= 2∜52/72

= 2βˆœπŸπŸ“/πŸ’πŸ—


  1. Classify the following in to like surds

(i) βˆšπŸπŸ’ , βˆšπŸπŸπŸ– , βˆšπŸ•πŸ“ , βˆšπŸ•πŸ , βˆšπŸ“πŸ’ , βˆšπŸπŸ’

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

= 32√3 , 23√2 , 5√3 , 3×2√2 , 3√(2×3), 2√(2×3)

= 9√3, 8√2, 5√3, 6√2, 3√6, 2√6

= {8√2, 6√2}, {9√3, 5√3}, {2√6, 3√6,}

={ βˆšπŸπŸπŸ–, βˆšπŸ•πŸ}, { βˆšπŸπŸ’πŸ‘, βˆšπŸ•πŸ“}, { βˆšπŸ“πŸ’, βˆšπŸπŸ’,}


(ii) βˆ›πŸπŸŽπŸŽπŸŽ, βˆ›πŸ”πŸ–πŸ”, βˆ›πŸ”πŸ’πŸ–, βˆ›πŸ‘πŸ•πŸ“, βˆ›πŸπŸπŸ–, βˆ›πŸπŸ’

= βˆ›(2Γ—52), βˆ›(73Γ—2), βˆ›(63Γ— 3), βˆ›(53Γ—3), βˆ›(43Γ—2), βˆ›(23Γ— 3)

= 10βˆ›2, 7βˆ›2, 6βˆ›3, 5βˆ›3, 4βˆ›2, 2βˆ›3

= {4βˆ›πŸ, 7βˆ›πŸ, 10βˆ›πŸ} & {2βˆ›πŸ‘, 5βˆ›πŸ‘, 6βˆ›πŸ‘}

= { βˆ›πŸπŸŽπŸŽπŸŽ, βˆ›πŸ”πŸ–πŸ” , βˆ›πŸπŸπŸ–} & { βˆ›πŸ”πŸ’πŸ–, βˆ›πŸ‘πŸ•πŸ“, βˆ›πŸπŸ’}


  1. Which of the following are pure surds?

(i) √296 = √(33Γ—37) = 2√(37Γ—2) = 2βˆšπŸ•πŸ’ – not pure surds

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

(iii) βˆ› 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) βˆ› 296 Β = βˆ›(23Γ—37) = 2 βˆ›πŸ‘πŸ• – 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) √(πŸπŸ“βˆš(πŸπŸ•))

= √27 = √33 = 3√3

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

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

= ∜(153 Γ— 27)

= ∜(225 Γ— 27)

= βˆœπŸ”πŸŽπŸ•πŸ“

Β 

(ii) √(πŸ’πŸŽ(√𝟏𝟐))

= [40 Γ— (12)1/2] Β½

= (40)1/2 Γ— (12)1/4

= ∜(402 Γ— (12)1)

= ∜(402Γ— 121)

= ∜(1600Γ—12)

= ∜19200

 

(iii) √(πŸ“(βˆšπŸ’πŸ–))

= [51/2 Γ—(48)1/2]Β½

= [51/2 Γ— 481/4]

= ∜ (52Γ— 48)

= ∜(25Γ—48)

= ∜1200


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

(i) βˆ› , √𝟐and πŸ“1/5

The orders are 3, 2, and 5

LCM of 2, 3 and 5 is 30

 

βˆ›2 = 21/3 = 210/30 = (𝟐𝟏𝟎) 1/3𝟎

√2 = 21/2 = 215/30 = (πŸπŸπŸ“ ) 1/3𝟎

51/5 = 51/5 = 56/30 = (πŸ“πŸ”) 1/3𝟎

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

They have the same order 30

 

(ii) βˆšπŸ“ , (βˆšπŸπŸ“)1/4 , and (βˆšπŸ“πŸŽ)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 Β , πŸπŸπŸ“πŸ– , πŸπŸπŸ“πŸ–

∴ Thus they all are of the same order 8.

 

(iii) √𝟐 , βˆšπŸ•1/3 , (√𝟏𝟏)1/4 and (βˆšπŸπŸ”πŸ•πŸ)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

πŸ”πŸ’πŸ/1𝟐 , πŸπŸ’πŸŽπŸπŸ/1𝟐 , πŸπŸ‘πŸ‘πŸπŸ/1𝟐 , πŸπŸ”πŸ•πŸπŸ/1𝟐

∴ Thus they all are of the same order 12


 

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