# ROOTS OF COMPLEX NUMBERS (Text)

$\color {royalblue} {Definition}:\ \hspace{20cm}$
$A\ number\ ω\ is\ called\ the\ n^{th}\ root\ of\ a\ complex\ number Z,$
$and\ we\ write\ ω = Z ^{\frac{1}{n}}$
$\color {royalblue} {Working\ rule\ to\ find\ the\ n^{th}\ roots\ of\ a\ complex numbers}:\ \hspace{18cm}$
$1)\ Write\ the\ given\ complex\ number\ in\ Polar\ form$
$2)\ Add\ ‘2kπ’\ to\ the\ argument$
$3)\ Apply\ De-Moivre’s\ theorem$
$4)\ Put\ k = 0,1,……. up to\ (n-1)$.
$\color {royalblue} {Illustration}:\ \hspace{20cm}$
$To\ find\ the\ n^{th} roots\ of\ unity\ or\ x^n\ -\ 1\ =\ 0\ \hspace{18cm}$
$\color {black}{Solution:}\ x^n\ -\ 1\ =\ 0 \hspace{18cm}$
$x^n\ =\ 1\ \hspace{18cm}$
$\ x\ =\ (1)^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{n}\ \hspace{10cm}$
$=\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{n}\ \hspace{10cm}$
$=\ cos\ (\frac{2kπ}{n})\ +\ i\ sin\ (\frac{2kπ}{n})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ ……..\ n-1\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{n}\ +\ i\ sin\ \frac{2π}{n}\ =\ ω$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{n}\ +\ i\ sin\ \frac{4π}{n}\ =\ ω^2$
$When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{n}\ +\ i\ sin\ \frac{6π}{n}\ =\ ω^3$
$.$
$.$

$When\ k = n-1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2(n-1)π}{n}\ +\ i\ sin\ \frac{2(n-1)π}{n}\ =\ ω^{n-1}$
$The\ n^{th}\ roots\ of\ unity\ are\ 1,\ ω,\ ω ^2,\ …………\ ω ^{n-1}$
$\color {royalblue} {Results}:\ \hspace{20cm}$
$1)\ If \ ω\ is\ n^{th}\ roots\ of\ unity,\ then\ \hspace{15cm}$
$( i )\ ω ^n\ =\ 1\ \hspace{10cm}$
$(ii)\ sum\ of\ roots\ is\ zero.\ \hspace{10cm}$
$i.e\ 1\ +\ ω\ +\ ω ^2\ +\ ………… +\ ω ^{n-1}= 0\ \hspace{10cm}$
$2)\ If\ ω\ is\ cube\ roots\ of\ unity,\ then\ \hspace{15cm}$
$( i )\ ω ^3\ =\ 1\ \hspace{10cm}$
$(ii)\ sum\ of\ roots\ is\ zero.\ \hspace{10cm}$
$i.e\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ \hspace{10cm}$
$\color {purple} {Example\ 1:}\ \color {red}{If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of} \ 1\ +\ ω\ +\ ω ^2\ \hspace{15cm}$
$\color {blue}{Solution:}\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ \hspace{18cm}$
$\color {purple} {Example\ 2:}\ \color {red} {If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of}\ ω ^3\ \hspace{18cm}$
$\color {blue}{Solution:}\ ω ^3\ =\ 1\ \hspace{20cm}$
$\color {purple} {Example\ 3:}\ \color {red} {If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of}\ ω ^4\ +\ ω ^5\ +\ ω ^6\ \hspace{16cm}$
$\color {blue}{Solution:}\ ω ^4\ +\ ω ^5\ +\ ω ^6\ =\ ω ^4( 1\ +\ ω\ +\ ω ^2 )\ =\ ω ^4 ( 0 )\ =\ 0\ \hspace{6cm}$
$\color {purple} {Example\ 4:}\ \color {red} {Solve}\ x^2\ -\ 1\ =\ 0\ \hspace{18cm}$
$\color {blue}{Solution:}\ x^2\ -\ 1\ =\ 0 \hspace{18cm}$
$x^2\ =\ 1\ \hspace{10cm}$
$\color {purple} {Example\ 5:}\ \color {red} {Find\ all\ the\ values\ of}\ (1)^\frac{1}{3}\ (OR)\ \color {red} {Solve}\ x^3\ -\ 1\ =\ 0\ \hspace{18cm}$
$\color {black}{Solution:}\ x\ =\ (1)^\frac{1}{3}\ \hspace{18cm}$
$=\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{3}\ \hspace{10cm}$
$=\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{3}\ \hspace{10cm}$
$=\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{3}\ \hspace{10cm}$
$=\ cos\ (\frac{2kπ}{3})\ +\ i\ sin\ (\frac{2kπ}{3})\ where\ k\ =\ 0,\ 1,\ 2\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{3}\ +\ i\ sin\ \frac{2π}{3}$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{3}\ +\ i\ sin\ \frac{4π}{3}$
$\color {purple} {Example\ 6:}\ \color {red} {Solve}\ x^6 -\ 1\ =\ 0\ \hspace{18cm}$
$\color {blue}{Solution:}\ x^6\ -\ 1\ =\ 0 \hspace{18cm}$
$x^6\ =\ 1\ \hspace{10cm}$
$\ x\ =\ (1)^\frac{1}{6}\ \hspace{10cm}$
$=\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{6}\ \hspace{10cm}$
$=\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{6}\ \hspace{10cm}$
$=\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{6}\ \hspace{10cm}$
$=\ cos\ (\frac{2kπ}{6})\ +\ i\ sin\ (\frac{2kπ}{6})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4,\ 5\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{6}\ +\ i\ sin\ \frac{2π}{6}\ =\ cos\ \frac{π}{3}\ +\ i\ sin\ \frac{π}{3}$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{6}\ +\ i\ sin\ \frac{4π}{6}\ =\ cos\ \frac{2π}{3}\ +\ i\ sin\ \frac{2π}{3}$
$When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{6π}{6}\ +\ i\ sin\ \frac{6π}{6}\ =\ cos\ π\ +\ i\ sin\ π$
$When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{8π}{6}\ +\ i\ sin\ \frac{8π}{6}\ =\ cos\ \frac{4π}{3}\ +\ i\ sin\ \frac{4π}{3}$
$When\ k = 5,\ \hspace{2cm}\ x\ =\ cos\ \frac{10π}{6}\ +\ i\ sin\ \frac{10π}{6}\ =\ cos\ \frac{5π}{3}\ +\ i\ sin\ \frac{5π}{3}$
$\color {purple} {Example\ 7:}\ \color {red} {Solve}\ x^5 + 1\ =\ 0\ \hspace{18cm}$
$\color {blue}{Solution:}\ x^5\ +\ 1\ =\ 0 \hspace{18cm}$
$x^5\ =\ -\ 1\ \hspace{10cm}$
$\ x\ =\ (-\ 1)^\frac{1}{5}\ \hspace{10cm}$
$=\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{5}\ \hspace{10cm}$
$=\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{5}\ \hspace{10cm}$
$=\ cos\ (\frac{π\ + 2kπ}{5})\ +\ i\ sin\ (\frac{π\ + 2kπ}{5})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{5}\ +\ i\ sin\ \frac{π}{5}$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{5}\ +\ i\ sin\ \frac{3π}{5}$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{5}\ +\ i\ sin\ \frac{5π}{5}\ =\ -1$
$When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{7π}{5}\ +\ i\ sin\ \frac{7π}{5}$
$When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{9π}{5}\ +\ i\ sin\ \frac{9π}{5}$
$\color {purple} {Example\ 8:}\ \color {red} {Solve}\ x^8\ -\ x^5\ +\ x^3\ -\ 1\ =\ 0\ \hspace{18cm}$
$\color {blue}{Solution:}\ Given\ x^8\ -\ x^5\ +\ x^3\ -\ 1\ =\ 0\ =\ 0\ \hspace{15cm}$
$x^5(x^3\ -\ 1)\ +\ 1( x^3\ -\ 1)\ =\ 0\ \hspace{10cm}$
$(x^5\ +\ 1)\ ( x^3\ -\ 1)\ =\ 0\ \hspace{10cm}$
$\color {green} {Case\ ( i ) :}\ \hspace{16cm}$
$x^5\ +1 =\ 0\ \hspace{10cm}$
$x^5\ =\ -\ 1\ \hspace{10cm}$
$\ x\ =\ (-\ 1)^\frac{1}{5}\ \hspace{10cm}$
$=\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{5}\ \hspace{10cm}$
$=\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{5}\ \hspace{10cm}$
$=\ cos\ (\frac{π\ + 2kπ}{5})\ +\ i\ sin\ (\frac{π\ + 2kπ}{5})\ where\ k\ =\ 0,\ 1,\ 2,\ 3,\ 4\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{5}\ +\ i\ sin\ \frac{π}{5}$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{5}\ +\ i\ sin\ \frac{3π}{5}$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{5}\ +\ i\ sin\ \frac{5π}{5}\ =\ -1$
$When\ k = 3,\ \hspace{2cm}\ x\ =\ cos\ \frac{7π}{5}\ +\ i\ sin\ \frac{7π}{5}$
$When\ k = 4,\ \hspace{2cm}\ x\ =\ cos\ \frac{9π}{5}\ +\ i\ sin\ \frac{9π}{5}$
$\color {green} {Case\ ( ii ) :}\ \hspace{16cm}$
$x^3\ -\ 1 =\ 0\ \hspace{10cm}$
$x^3\ =\ 1\ \hspace{10cm}$
$\ x\ =\ (1)^\frac{1}{3}\ \hspace{10cm}$
$=\ (cos\ 0\ +\ i\ sin\ 0)^\frac{1}{3}\ \hspace{10cm}$
$=\ (cos\ (0\ + 2kπ) +\ i\ sin\ (0\ + 2kπ))^\frac{1}{3}\ \hspace{10cm}$
$=\ (cos\ 2kπ\ +\ i\ sin\ 2kπ)^\frac{1}{3}\ \hspace{10cm}$
$=\ cos\ (\frac{2kπ}{3})\ +\ i\ sin\ (\frac{2kπ}{3})\ where\ k\ =\ 0,\ 1,\ 2\ \hspace{5cm}$
$The\ roots\ are$
$When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ 0\ +\ i\ sin\ 0\ =\ 1$
$When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{2π}{3}\ +\ i\ sin\ \frac{2π}{3}$
$When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{4π}{3}\ +\ i\ sin\ \frac{4π}{3}$

### Exercise Problems

$\LARGE{\color {purple} {PART- A}}$
$\color {purple} {1.}\ \color {red} {If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of}\ ω( ω\ +\ 1)\ \hspace{15cm}$
$\LARGE{\color {purple} {PART- C}}$
$\color {purple} {2\ .}\ \color {red} {Solve}\ x^4 -\ 1\ =\ 0\ \hspace{18cm}$
$\color {purple} {3:}\ \color {red} {Solve}\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{18cm}$