**Find which is larger:**

**(i) 3****∛**𝟑 **and 4****∜**𝟒

**∛(**3×3^{2}) 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

**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} = (30^{15})^{1/420 }

28^{14} =2^{2} x 7^{14} = 2^{28} x 7^{14}

30^{15} = (5 x 6)^{15} = 5^{15} x 6^{15 }

Comparing the 2 numbers we conclude

30^{15} > 28^{14}

(∜(30))^{1/7} > ∛(28^{1/10})

**(ii) √(∜8) ** **and ∛(∛9)**

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

∛(∛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} = (2^{27})^{1/72}

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

By comparing we find that (2^{27}) is larger than 3^{16}.

**Hence **√(∜8) **> **∛(∛9)

**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} = (2^{3})^{1/6} = 8^{1/6}

3^{1/3} = 3^{2/6} = (3^{3})^{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}

**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}

(6^{4})^{1/24}, (12^{2})^{1/24}, (8^{3})^{1/24}

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

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

Pingback: IX – Table of Contents – Breath Math