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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 \dots

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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