A) \[\sqrt[9]{4},\sqrt[6]{3},\sqrt[3]{2}\]
B) \[\sqrt[9]{4},\sqrt[3]{2},\sqrt[6]{3}\]
C) \[\sqrt[3]{2},\sqrt[6]{3},\sqrt[9]{4}\]
D) \[\sqrt[6]{3},\sqrt[9]{4},\sqrt[3]{2}\]
Correct Answer: A
Solution :
LCM of 3, 6, 9 = 18 \[\therefore \]\[\sqrt[3]{2}={{(2)}^{\frac{1}{3}\times \frac{6}{6}}}={{({{2}^{6}})}^{\frac{1}{18}}}={{(64)}^{\frac{1}{18}}}=\sqrt[18]{64}\] \[\sqrt[6]{3}={{(3)}^{\frac{1}{6}\times \frac{3}{3}}}={{({{3}^{3}})}^{\frac{1}{18}}}={{(27)}^{\frac{1}{18}}}=\sqrt[18]{27}\] \[\sqrt[9]{4}={{(4)}^{\frac{1}{9}\times \frac{2}{2}}}={{({{4}^{2}})}^{\frac{1}{18}}}={{(16)}^{\frac{1}{18}}}=\sqrt[18]{16}\] So, the order is, \[\sqrt[18]{27}<\sqrt[18]{27}<\sqrt[18]{64}\] \[\Rightarrow \]\[\sqrt[9]{4}<\sqrt[18]{27}<\sqrt[18]{64}\]
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