ROOTS OF COMPLEX NUMBERS (REVISION)

\[1.\ Solve\ x^5 -\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ x^5\ -\ 1\ =\ 0 \hspace{18cm}\]
\[ x^5\ =\ 1\ \hspace{10cm}\]
\[\ x\ =\ (1)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{2kπ}{5})\ +\ i\ sin\ (\frac{2kπ}{5})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4,\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{5}\ +\ i\ sin\ \frac{2π}{5}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{5}\ +\ i\ sin\ \frac{4π}{5}\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{5}\ +\ i\ sin\ \frac{6π}{5}\]
\[When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{8π}{5}\ +\ i\ sin\ \frac{8π}{5}\]
\[2.\ Solve\ x^9 +\ x^5\ -\ x^4\ -\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Given\ x^9 +\ x^5\ -\ x^4\ -\ 1\ =\ 0\ \hspace{15cm}\]
\[x^5(x^4 +\ 1)\ -\ 1( x^4\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[(x^5\ -\ 1)\ ( x^4\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[Case\ ( i ) :\ \hspace{16cm}\]
\[x^5\ -\ 1 =\ 0\ \hspace{10cm}\]
\[ x^5\ =\ 1\ \hspace{10cm}\]
\[\ x\ =\ (1)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{5}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{2kπ}{5})\ +\ i\ sin\ (\frac{2kπ}{5})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4,\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{5}\ +\ i\ sin\ \frac{2π}{5}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{5}\ +\ i\ sin\ \frac{4π}{5}\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{5}\ +\ i\ sin\ \frac{6π}{5}\]
\[When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{8π}{5}\ +\ i\ sin\ \frac{8π}{5}\]
\[Case\ ( ii ) :\ \hspace{16cm}\]
\[x^4\ +1 =\ 0\ \hspace{10cm}\]
\[ x^4\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{4}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{4}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{4}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{4})\ +\ i\ sin\ (\frac{π\ + 2kπ}{4})\ where\ k\ =\ 0,\ 1,\ 2,\ 3\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{4}\ +\ i\ sin\ \frac{π}{4}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{4}\ +\ i\ sin\ \frac{3π}{4}\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{4}\ +\ i\ sin\ \frac{5π}{4}\]
\[When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{7π}{4}\ +\ i\ sin\ \frac{7π}{4}\]