Question

find the continued product of all the values of (1 + i)^(1/5)​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{(1+i)^\dfrac{1}{5}}$

$\underline{\textbf{To find:}}$

$\textsf{Continued product of all the values of}\;\mathsf{(1+i)^\dfrac{1}{5}}$

$\underline{\textbf{Solution:}}$

$\textsf{First we write 1+i in polar form}$

$\mathsf{1+i=\sqrt2\left(\dfrac{1}{\sqrt2}+i\dfrac{1}{\sqrt2}\right)}$

$\implies\mathsf{1+i=\sqrt2\left(cos\dfrac{\pi}{4}+i\,sin\dfrac{\pi}{4}\right)}$

$\mathsf{Add\;2k\pi\;to\;argument}$

$\implies\mathsf{1+i=\sqrt2\left(cos\left(2k\pi+\dfrac{\pi}{4}\right)+i\,sin\left(2k\pi+\dfrac{\pi}{4}\right)\right)}$

$\implies\mathsf{(1+i)^{\dfrac{1}{5}}=\left[\sqrt2\left(cos\left(2k\pi+\dfrac{\pi}{4}\right)+i\,sin\left(2k\pi+\dfrac{\pi}{4}\right)\right)\right]^\dfrac{1}{5}}$

$\implies\mathsf{(1+i)^{\dfrac{1}{5}}=\left[\sqrt2\left(cos\left(\dfrac{8k\pi+\pi}{4}\right)+i\,sin\left(\dfrac{8k\pi+\pi}{4}\right)\right)\right]^\dfrac{1}{5}}$

$\textsf{Using Demovire's theorem}$

$\implies\mathsf{(1+i)^{\dfrac{1}{5}}={\sqrt2}^{\dfrac{1}{5}}\left(cos\dfrac{1}{5}\left(\dfrac{8k\pi+\pi}{4}\right)+i\,sin\dfrac{1}{5}\left(\dfrac{8k\pi+\pi}{4}\right)\right)\right}\;\;\;\;\mathsf{k=0,1,2,3,4}$

$\implies\mathsf{(1+i)^{\dfrac{1}{5}}={\sqrt2}^{\dfrac{1}{5}}\left(cos\left(\dfrac{8k\pi+\pi}{20}\right)+i\,sin\left(\dfrac{8k\pi+\pi}{20}\right)\right)\right}\;\;\;\;\mathsf{k=0,1,2,3,4}$

$\implies\mathsf{(1+i)^{\dfrac{1}{5}}={\sqrt2}^{\dfrac{1}{5}}\,e^{i\left(\dfrac{8k\pi+\pi}{20}}\right)}\;\;\;\;\mathsf{k=0,1,2,3,4}$

$\therefore\textsf{The values are}$

$\mathsf{For\;k=0,\;{\sqrt2}^{\dfrac{1}{5}}\,e^{i\,\dfrac{\pi}{20}}}$

$\mathsf{For\;k=1,\;{\sqrt2}^{\dfrac{1}{5}}\,e^{i\,\dfrac{9\pi}{20}}}$

$\mathsf{For\;k=2,\;{\sqrt2}^{\dfrac{1}{5}}\\,e^{i\,\dfrac{17\pi}{20}}}$

$\mathsf{For\;k=3,\;{\sqrt2}^{\dfrac{1}{5}}\,e^{\dfrac{i\,25\pi}{20}}}$

$\mathsf{For\;k=4,\;{\sqrt2}^{\dfrac{1}{5}}\,e^{i\,\dfrac{33\pi}{20}}}$

$\mathsf{Now,}$

$\textsf{Product of all the values}$

$\mathsf{={\sqrt2}^{\dfrac{1}{5}}\,e^{i\dfrac{\pi}{20}}.{\sqrt2}^{\dfrac{1}{5}}\,e^{i\dfrac{9\pi}{20}}.{\sqrt2}^{\dfrac{1}{5}}\,e^{i\dfrac{17\pi}{20}}.{\sqrt2}^{\dfrac{1}{5}}\,e^{i\dfrac{25\pi}{20}}.{\sqrt2}^{\dfrac{1}{5}}\,e^{i\dfrac{i\,33\pi}{20}}}$

$\mathsf{=\sqrt2\,e^{i\dfrac{85\pi}{20}}}$

$\mathsf{={\sqrt2}\,e^{i\dfrac{17\pi}{4}}}$

$\mathsf{={\sqrt2}\left(cos\dfrac{17\pi}{4}+i\,sin\dfrac{17\pi}{4}\right)}$

$\mathsf{={\sqrt2}\left(cos\left(4\pi+\dfrac{\pi}{4}\right)+i\,sin\left(4\pi+\dfrac{\pi}{4}\right)\right)}$

$\mathsf{={\sqrt2}\left(cos\left(\dfrac{\pi}{4}\right)+i\,sin\left(\dfrac{\pi}{4}\right)\right)}$

$\mathsf{={\sqrt2}\left(\dfrac{1}{\sqrt2}+i\,\dfrac{1}{\sqrt2}\right)}$

$\mathsf{=1+i}$