Question

If $2^{\frac{1}{4}}{.4}^{\frac{1}{8}}{.8}^{\frac{1}{16}}.{16}^{\frac{1}{32}}...............={32}^{\frac{1}{m}}$. Find $m$

Answer 1

$2^{\frac{1}{4}}.4^{\frac{1}{8}}.8^{\frac{1}{16}}....={32}^{\frac{1}{m}}{2}^{\frac{1}{4}}.2^{\frac{1}{4}}{.4}^{\frac{3}{16}}....=2^{\frac{5}{m}}\frac{1}{4}+\frac{1}{4}+\frac{3}{16}+\frac{1}{8}+....=\frac{5}{m}\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+....=\frac{5}{m}\frac{5}{m}-\frac{5}{2m}=(\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}....)-(\frac{1}{8}+\frac{2}{16}+\frac{3}{32})\frac{5}{2m}=(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}....)\frac{5}{2m}=\frac{\left(\frac{1}{4}\right)}{(1-\frac{1}{2})}.\frac{5}{2m}=\frac{1}{2}m=5$