Which of the following is TRUE \({\rm{I}}.{\rm{\;}}\sqrt[3]{{11}} > \sqrt 7 > \sqrt[4]{{45}}\) \({\rm{II}}.{\rm{\;}}\sqrt 7 > \sqrt[3]{{11}} > \sqrt[4]{{45}}\) \({\rm{III}}.{\rm{\;}}\sqrt 7 > \sqrt[4]{{45}} > \sqrt[3]{{11}}\) \({\rm{IV}}.{\rm{\;}}\sqrt[4]{{45}} > \sqrt 7 > \sqrt[3]{{11}}\)?

Option 3 : Only III

Taking power of 12 in all the terms;

I. 11⁴ > 7⁶ > 45³ = 14641 > 117649 > 91125

II. 7⁶ > 11⁴ > 45³ = 117649 > 14641 > 91125

III. 7⁶ > 45³ > 11⁴ = 117649 > 91125 > 14641

IV. 45³ > 7⁶ > 11⁴ = 91125 > 117649 > 14641

Here we can see that statement III is true only

Alternate Method

(11)^1/3, (7)^1/2 and 45^1/4

Take LCM of (3, 2 and 4) = 12 

⇒ (11)^1/3 = (11⁴)^1/12 = (14641)^1/12

⇒ (7)^1/2 = (7⁶)^1/12 = (117649)^1/12

⇒ (45)^1/4 = (45³)^1/12 = (91125)^1/12

So, Correct order = (117649)^1/12 > (91125)^1/12 >  (14641)^1/12

⇒ 7^1/2 > 45^1/4 > 11^1/3 

∴ The correct answer is statement III.