| The greatest among the numbers \[\sqrt[4]{2},\]\[\sqrt[5]{3},\]\[\sqrt[10]{6},\]\[\sqrt[20]{15}\]is [SSC (CGL) 2014] |
A) \[\sqrt[20]{15}\]
B) \[\sqrt[4]{2}\]
C) \[\sqrt[5]{3}\]
D) \[\sqrt[10]{6}\]
Correct Answer: C
Solution :
| In \[\sqrt[x]{a},\]\[a\]is called base and x is called the radical. |
| Now, first take the LCM of 4, 5, 10 and 20. |
| LCM of 4, 5, 10 and \[20=20\] |
| \[\sqrt[4]{2}={{(2)}^{\frac{1}{4}}}={{({{2}^{5}})}^{\frac{1}{20}}}={{(32)}^{\frac{1}{20}}}\]\[[\because \sqrt[x]{y}={{y}^{1/x}}]\] |
| \[\sqrt[5]{3}={{(3)}^{\frac{1}{5}}}={{({{3}^{4}})}^{\frac{1}{20}}}={{(81)}^{\frac{1}{20}}}\] |
| \[\sqrt[10]{6}={{(6)}^{\frac{1}{10}}}={{({{6}^{2}})}^{\frac{1}{20}}}={{(36)}^{\frac{1}{20}}}\] |
| \[\sqrt[20]{15}={{(15)}^{\frac{1}{20}}}\] |
| Thus, greatest number is \[{{(81)}^{\frac{1}{20}}},\]i.e. \[\sqrt[5]{3}.\] |
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