Please, help i cannot solve problems such as: x^4 =1, x^5 =1, or x^n =1 n∈N...


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Question Number 101921 by Farruxjano last updated on 05/Jul/20

$P l e a s e, h e l p i c a n n o t s o l v e p r o b l e m s$ $s u c h a s : x^{4} = 1, x^{5} = 1, o r x^{n} = 1 n \in N . . .$
Commented by prakash jain last updated on 05/Jul/20

$Search internet for roots of unity .$ $You will find lot of information .$ $to start with i = \sqrt{- 1}$ $e^{i θ} = \cos θ + i \sin θ$ $1 = \cos 2 π k + i \sin 2 π k$ $1^{1 / n} = e^{\frac{2 π j}{n}} (j = 0, . . . . n - 1)$ $as you can see {(e^{\frac{2 π j}{n}})}^{n} = e^{2 j π} = 1$
	Answered by 1549442205 last updated on 06/Jul/20
	$Applying the Mauvra′s formular (cosϕ+isinϕ)^n =cosnϕ+isinnϕ Since 1=cos2kπ+isin2kπ ,so i)For the eqs.x^4 =1 we get x=1^(1/4) ⇔x=(cos2kπ+isin2kπ)^(1/4) =cos((2kπ)/4)+isin((2kπ)/4)(k=0,1,2,3) we get four different roots ii)For the eqs.x^5 =1⇔x=1^(1/5) .We get x=(cos2kπ+isin2kπ)^(1/5) ⇔x=cos((2kπ)/5)+isin((2kπ)/5)(k∈{0,1,2,3,4}) weget five different roots iii)For eqs.x^n =1⇔x=1^(1/n) .We get x=(cos2kπ+isin2kπ)^(1/n) =cos((2kπ)/n)+isin((2kπ)/n)(n∈{0,1,2,...n−1}) weget n different roots for k=n,n+1,n+2,... we get the repeated roots$
	$Applying the {Mauvra}^{'} s formular$ ${(\cos φ + isin φ)}^{n} = cosn φ + isinn φ$ $Since 1 = \cos 2 k π + isin 2 k π, so$ $i) For the eqs . x^{4} = 1 we get x = 1^{\frac{1}{4}}$ $\Leftrightarrow x = {(\cos 2 k π + isin 2 k π)}^{\frac{1}{4}} = \cos \frac{2 k π}{4} + isin \frac{2 k π}{4} (k = 0, 1, 2, 3)$ $we get four different roots$ $ii) For the eqs . x^{5} = 1 \Leftrightarrow x = 1^{\frac{1}{5}} . We get$ $x = {(\cos 2 k π + isin 2 k π)}^{\frac{1}{5}} \Leftrightarrow x = \cos \frac{2 k π}{5} + isin \frac{2 k π}{5} (k \in {0, 1, 2, 3, 4})$ $weget five different roots$ $iii) For eqs . x^{n} = 1 \Leftrightarrow x = 1^{\frac{1}{n}} . We get$ $x = {(\cos 2 k π + isin 2 k π)}^{\frac{1}{n}} = \cos \frac{2 k π}{n} + isin \frac{2 k π}{n} (n \in {0, 1, 2, . . . n - 1})$ $weget n different roots$ $for k = n, n + 1, n + 2, . . . we get the repeated roots$
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