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Which of the following is TRUE? \({\rm{I}}.{\rm{\;}}\frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt 5 }}\) \({\rm{II}}.\;\frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt 5 }}\) \({\rm{III}}.\;\frac{1}{{\sqrt 5 }} > \frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt[4]{{29}}}}\) \({\rm{IV}}.{\rm{\;}}\frac{1}{{\sqrt 5 }} > \frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt[3]{{12}}}}\)

Which of the following is TRUE? \({\rm{I}}.{\rm{\;}}\frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt 5 }}\) \({\rm{II}}.\;\frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt 5 }}\) \({\rm{III}}.\;\frac{1}{{\sqrt 5 }} > \frac{1}{{\sqrt[3]{{12}}}} > \frac{1}{{\sqrt[4]{{29}}}}\) \({\rm{IV}}.{\rm{\;}}\frac{1}{{\sqrt 5 }} > \frac{1}{{\sqrt[4]{{29}}}} > \frac{1}{{\sqrt[3]{{12}}}}\) Correct Answer Only III

Calculation:

1/∛12, 1/4√29, and 1/√5   

The powers are 3, 4 and 1 

LCM of 3, 4, and 1 is 12

Taking the power of 12 on all three numbers, we get

⇒ (1/12)4, (1/29)3 and (1/5)6

⇒ (1/144)2, 1/(841 × 29), and (1/125)2

Now we can say that,

⇒ (1/125)2 > (1/144)2 > 1/(841 × 29)

⇒ 1/√5 > 1/∛12 > 1/4√29

∴ Option 3 is correct.

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