ABC26GN1993 · Surds and Indices

L.C.M of 2, 3, 4, 6 is 12. $$\begin{aligned} & \sqrt{2}=2^{\frac{1}{2}}=2^{\left(\frac{1}{2} \times \frac{6}{6}\right)}=2^{\frac{6}{12}}=\left(2^{6}\right)^{\frac{1}{12}}=(64)^{\frac{1}{12}}=\sqrt[12]{64} . \\ & \sqrt[6]{3}=3^{\frac{1}{6}}=3^{\left(\frac{1}{6} \times \frac{2}{2}\right)}=3^{\frac{2}{12}}=\left(3^{2}\right)^{\frac{1}{12}}=(9)^{\frac{1}{12}}=\sqrt[12]{9} . \\ & \sqrt[3]{4}=4^{\frac{1}{3}}=4^{\left(\frac{1}{3} \times \frac{4}{4}\right)}=4^{\frac{4}{12}}=\left(4^{4}\right)^{\frac{1}{12}}=(256)^{\frac{1}{12}}=\sqrt[12]{256} . \\ & \sqrt[4]{5}=5^{\frac{1}{4}}=5^{\left(\frac{1}{4} \times \frac{3}{3}\right)}=5^{\frac{3}{12}}=\left(5^{3}\right)^{\frac{1}{12}}=(125)^{\frac{1}{12}}=\sqrt[12]{125} . \end{aligned}$$ Clearly, $\sqrt[12]{256}$ i.e., $\sqrt[3]{4}$ is the greatest.