SURDS EXERCISE – 1.3.4 – Class 9

1. Find which is larger:

(i) 3∛𝟑 and 4∜𝟒

∛(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

4∜𝟒 is greater 3∛𝟑

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2. Compare the following and decide which is larger.

(i) (∜(30))^1/7 and ∛(28^1/10)

(28^1/10) ^1/3 = (28^1/10)^1/3 = 28^1/30

(30^1/4) ^1/7 = (30^1/4)^1/7 = 30^1/28

LCM of 30 and 28 is 420

(28^1/10)^1/3 = 28^1/30 = (28^1/14)^1/420

(30^1/4) ^1/7 = 30^1/28 = (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 > ∛(28^1/10)

 

(ii) √(∜8)  and ∛(∛9)

√(∜8) = (8^1/4)^1/2 = 8^1/8 = 2^3/8          [8 = 2³]

∛(∛9) = (9^1/3)^1/3 = (3^2/3)^1/3 = 3^2/9

LCM of 8 and 9 is 72

√(∜8)  = 2^3/8 = (2^3/8)^9/9 = 2^27/72 = (2²⁷)^1/72

∛(∛9) = 3^2/9 = (3^2/9)^8/8 = 3^16/72 = (3¹⁶)^1/72

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

Hence √(∜8)   > ∛(∛9)

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3. Write the following in ascending order:

√𝟐 , ∛𝟑𝟑 , 𝟔^1/6

2^1/2 , 3^1/3 , 6^1/6

LCM of 2, 3 and 6 is 6

2^1/2 = 2^3/6 = (2³)^1/6 = 8^1/6

3^1/3 = 3^2/6 = (3³)^1/6 = 9^1/6

6^1/6 = 6^3/6 = (6)^1/6 = 6^1/6

Ascending order is 6^1/6  , 2^1/2, 3^1/3

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4. Write the following descending order:

√(∛𝟔) , ∛(∜𝟏𝟐) , √ (∜𝟖)

(6^1/3)^1/2 , (12^1/4)^1/3 , (8^1/4)^1/2

6^1/6, 12^1/12 , 8^1/8

LCM of 6, 12, 8 is 24

6^4/24, 12^2/24, 8^3/24

(6⁴)^1/24, (12²)^1/24, (8³)^1/24

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

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

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