x^(3z) =1 x^2 =y z=y^n FIND THE VALUE OF n please i need your help ASAP. thanks


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Question Number 34056 by Learner last updated on 30/Apr/18

$x^{3 z} = 1$ $x^{2} = y$ $z = y^{n}$ $F I N D T H E V A L U E O F n$ $p l e a s e i n e e d y o u r h e l p A S A P . t h a n k s$
Commented by math1967 last updated on 30/Apr/18

$x^{3 z} = 1 \Rightarrow x^{3 z} = x^{0} ∴ z = 0$ $y^{n} = z \Rightarrow y^{n} = 0 \Rightarrow n o w h o w c a n I f i n d$ $n ? ? ? ? ? ? ? ? ?$
Commented by Learner last updated on 30/Apr/18

$p l e a s e s i r h o w z = 0 ? ? ? ? ?$ $i s i t t h a t t h e r e i s n o v a l u e o f n t o g i v e a s o l u t i o n t o t h e e q u a t i o n ? ? ? ?$
Commented by MJS last updated on 30/Apr/18

${it}^{'} s only 3 equations in 4 unknown numbers . . .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 30/Apr/18

	$x^{3 z} = 1^{3 z}$ $x = 1$ $y = 1^{2} = 1$ $z = 1$ $o r$ $x^{3 z} = 1 = x^{0}$ $z = 0$ $y^{n} = z$ $y^{n} = 0$ $n = - \infty$
	Commented by tanmay.chaudhury50@gmail.com last updated on 30/Apr/18

	$w h e n y n o t e q u a l s t o 1$
	Answered by MJS last updated on 30/Apr/18

	$if n, x, y, z \in Z (if {they}^{'} re \in C {it}^{'} s getting complicated)$ $x^{3 z} = 1 \Rightarrow$ $\Rightarrow (x = - 1 \land z = 2 k \land k \in N) \lor (x = 1 \land z \in Z) \lor (z = 0 \land x \in Z)$ $1 . x = - 1 \land z = 2 k \land k \in N$ $y = x^{2} = 1$ $z = y^{n} = 1 but 1 \neq 2 k with k \in N \Rightarrow no solution$ $2 . x = 1 \land z \in Z$ $y = x^{2} = 1$ $z = y^{n} = 1$ $solution : x = y = z = 1 \land n \in Z$ $3 . z = 0 \land x \in Z$ $z = y^{n} = 0 \Rightarrow y = 0 \land n \in Z$ $y = x^{2} = 0 \Rightarrow x = 0$ $solution x = y = z = 0 \land n \in Z (if we define 0^{0} = 1)$
	Commented by MJS last updated on 30/Apr/18

	$easy complex solution :$ $n = 0$ $z = y^{n} = 1$ $x^{3} = 1 \Rightarrow x = 1 \lor x = - \frac{1}{2} - \frac{\sqrt{3}}{2} i \lor x = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ $y = x^{2} \Rightarrow y = 1 \lor y = - \frac{1}{2} + \frac{\sqrt{3}}{2} i \lor y = - \frac{1}{2} - \frac{\sqrt{3}}{2} i$ $solutions :$ $(n ∣ x ∣ y ∣ z) = (0 ∣ 1 ∣ 1 ∣ 1) \lor (0 ∣ - \frac{1}{2} - \frac{\sqrt{3}}{2} i ∣ - \frac{1}{2} + \frac{\sqrt{3}}{2} i ∣ 1) \lor (0 ∣ - \frac{1}{2} + \frac{\sqrt{3}}{2} i ∣ - \frac{1}{2} - \frac{\sqrt{3}}{2} i ∣ 1)$
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