Question 54
From among $$2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 8^{\frac{1}{8}}$$ and $$9^{\frac{1}{9}}$$, the greatest is
Solution
Terms : $$2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 8^{\frac{1}{8}}$$ and $$9^{\frac{1}{9}}$$
L.C.M. (2,3,8,9) = 72
Multiplying the exponents by 72, we get :
= $$2^{\frac{72}{2}}, 3^{\frac{72}{3}}, 8^{\frac{72}{8}}$$ and $$9^{\frac{72}{9}}$$
= $$(2)^{36},(3)^{24},(8)^9,(9)^8$$
Now, clearly, $$(2)^{36}>(8)^9$$ and $$(3)^{24}>(9)^8$$
Now, among $$(2)^{36}$$ and $$(3)^{24}$$
these can be written as : $$(8)^{12}$$ $$<$$ $$(9)^{24}$$
Thus, greatest among these are : $$3^{\frac{1}{3}}$$
=> Ans - (B)
