Find the smallest among the following ? sqrt[4]6 ,sqrt 2, sqrt[3]4 an…

\(\sqrt[4]{6 },\sqrt { 2}, \sqrt[3]{4 }\) and \(\sqrt {3}\)

LCM of 4, 2, 3 is 12, now try to make the power same i.e.:

⇒ \(\sqrt[4]{6}\) = (6)^{\(\frac{1}{4}\)} =  (6)^{\(\frac{4}{12}\)} = (6⁴)^{\(\frac{1}{12}\)} =  (1296)^{\(\frac{1}{12}\)}

⇒ \(\sqrt {2}\) = (2)^{\(\frac{1}{2}\)} =  (2)^{\(\frac{6}{12}\)} = (2⁶)^{\(\frac{1}{12}\)} =  (64)^{\(\frac{1}{12}\)}

⇒ \(\sqrt[3]{4}\) = (4)^{\(\frac{1}{3}\)} =  (4)^{\(\frac{4}{12}\)} = (4⁴)^{\(\frac{1}{12}\)} = (256)^{\(\frac{1}{12}\)}

⇒ \(\sqrt {3}\) = (3)^{\(\frac{1}{2}\)} =  (3)^{\(\frac{6}{12}\)} = (3⁶)^{\(\frac{1}{12}\)} =  (729)^{\(\frac{1}{12}\)}

Hence,

⇒ (64)^{\(\frac{1}{12}\)} < (256)^{\(\frac{1}{12}\)} < (729)^{\(\frac{1}{12}\)} <  (1296)^{\(\frac{1}{12}\)}

⇒ \(\sqrt {2}\) < \(\sqrt[3]{4}\) < \(\sqrt {3}\) < \(\sqrt[4]{6}\)

Therefore, smallest no. is = \(\sqrt {2}\)