Which of the following is TRUE? \({\rm{I}}.{\rm{\;}}\sqrt[3]{{11}} > \sqrt 7 > \sqrt[4]{{45}}\) \({\rm{II}}.{\rm{\;}}\sqrt 7 > \sqrt[3]{{11}} > \sqrt[4]{{45}}\) \({\rm{III}}.{\rm{\;}}\sqrt 7 > \sqrt[4]{{45}} > \sqrt[3]{{11}}\) \({\rm{IV}}.{\rm{\;}}\sqrt[4]{{45}} > \sqrt 7 > \sqrt[3]{{11}}\)
Only III
Raise each radical to the LCM power 12:
\[(\sqrt[3]{11})^{12}=11^4=14641,\ (\sqrt7)^{12}=7^6=117649,\ (\sqrt[4]{45})^{12}=45^3=91125\]
Order: \(117649>91125>14641\), hence \(\sqrt7>\sqrt[4]{45}>\sqrt[3]{11}\).
This matches statement III.
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