1-If-is-an-imaginary-fifth-root-of-unity-then-find-value-of-log-2-1-2-3-1-2-Find-value-of-i-3-100-i-3-100-2-100-

Question Number 49202 by rahul 19 last updated on 04/Dec/18

1) I f ω i s a n i m a g i n a r y f i f t h r o o t o f

u n i t y, t h e n f i n d v a l u e o f

\log_{2} ∣ 1 + ω + ω^{2} + ω^{3} - \frac{1}{ω} ∣ ?

2) F i n d v a l u e o f :

{(i + \sqrt{3})}^{100} + {(i - \sqrt{3})}^{100} + 2^{100} ?

Commented by Abdo msup. last updated on 04/Dec/18

1) w^{5} = 1 \Rightarrow w^{5} - 1 = 0 \Rightarrow (w - 1) (1 + w + w^{2} + w^{3} + w^{4}) = 0 b u t w \in i R

\Rightarrow 1 + w + w^{2} + w^{3} + w^{4} = 0 \Rightarrow 1 + w + w^{2} + w^{3} = - w^{4} \Rightarrow

1 + w + w^{2} + w^{3} - \frac{1}{w} = - (w^{4} + \frac{1}{w}) = - \frac{w^{5} + 1}{w} = - \frac{2}{w} \Rightarrow

∣ 1 + w + w^{2} + w^{3} - \frac{1}{w} ∣= \frac{2}{∣ w ∣} = \frac{2}{1} = 2 \Rightarrow

{l o g}_{2} ∣ 1 + w + w^{2} + w^{3} - \frac{1}{w} ∣= {l o g}_{2} (2) = 1 .

Commented by Abdo msup. last updated on 04/Dec/18

2) w e h a v e i + \sqrt{3} = 2 (\frac{\sqrt{3}}{2} + \frac{i}{2}) = 2 e^{i \frac{π}{6}} a n d

i - \sqrt{3} = - (\sqrt{3} - i) = - 2 e^{- \frac{i π}{6}} \Rightarrow

{(\sqrt{3} + i)}^{100} + {(- \sqrt{3} + i)}^{100} + 2^{100}

= 2^{100} e^{i \frac{100 π}{6}} + 2^{100} e^{- i \frac{100 π}{6}} + 2^{100}

= 2^{101} c o s (\frac{100 π}{6}) + 2^{100} b u t

c o s (\frac{100 π}{6}) = c o s (\frac{50 π}{3}) = c o s (\frac{48 π + 2 π}{3}) = c o s (16 π + \frac{2 π}{3})

= c o s (\frac{2 π}{3}) = - \frac{1}{2} \Rightarrow {(\sqrt{3} + i)}^{100} + {(- \sqrt{3} + i)}^{100} + 2^{100}

= 2^{101} . (- \frac{1}{2}) + 2^{100} = - 2^{100} + 2^{100} = 0 .

Commented by rahul 19 last updated on 05/Dec/18

thank you prof Abdo

Commented by maxmathsup by imad last updated on 05/Dec/18

y o u a r e w e l c o m e

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Dec/18

2) \sqrt{3} + i

= 2 (\frac{\sqrt{3}}{2} + i \frac{1}{2}) = 2 (c o s \frac{π}{6} + i s i n \frac{π}{6}) = 2 e^{i \frac{π}{6}}

i - \sqrt{3}

= - 2 (\frac{\sqrt{3}}{2} - i \frac{1}{2}) = - 2 e^{- i \frac{π}{6}}

{(2 e^{i \frac{π}{6}})}^{100} + {(- 2 e^{- i \frac{π}{6}})}^{100} + 2^{100}

= 2^{100} {e^{i \frac{100 π}{6}} + e^{- i \frac{100 π}{6}}} + 2^{100}

= 2^{100} (2 c o s \frac{100 π}{6}) + 2^{100}

c o s (3000^{o}) = c o s (8 \times 360^{o} + 120^{o}) = c o s (120^{o})

c o s (120^{o}) = \frac{- 1}{2}

s o a n s i s

2^{100} \times 2 \times \frac{- 1}{2} + 2^{100} = 0

Commented by rahul 19 last updated on 05/Dec/18

thank you sir! ��

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Dec/18

1) x^{5} = 1

(x - 1) (x^{4} + x^{3} + x^{2} + x + 1) = 0

w^{4} + w^{3} + w^{2} + w + 1 = 0

1 + w + w^{2} + w^{3} - \frac{1}{w}

= - w^{4} - \frac{1}{w}

= \frac{- w^{5} - 1}{w} = \frac{- 1 - 1}{1} = - 2

s o ∣ 1 + w + w^{2} + w^{3} - \frac{1}{w} ∣=∣ - 2 ∣= 2

h e n c e {l n}_{2} 2 = 1