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Algebra Level pending

Given that x = y z x = y^z , y = z x y = z^x , and z = x y z = x^y , find the value of

( x y z ) ( x y z ) x y z \Large (xyz)^{(xyz)^{xyz}}

The problem is not original.


The answer is 1.

Razing Thunder by Razing Thunder , 16, India

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2 solutions

Chew-Seong Cheong
May 14, 2021

Given that

x = y z and y = z x = z x z and z = x y = x x y z ⟹ x = x x y z ⟹ x y z = 1 ⟹ ( x y z ) ( x y z ) x y z = 1 1 1 = 1 \large \begin{aligned} x & = \blue y^z & \small \blue{\text{and }y=z^x} \\ & = \blue z^{\blue xz} & \small \blue{\text{and }z=x^y} \\ & = \blue x^{x\blue yz} \\ \implies x & = x^{xyz} \\ \implies xyz & = 1 \\ \implies (xyz)^{(xyz)^{xyz}} & = 1^{1^1} = \boxed 1 \end{aligned}

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When you said xyz equals 1 because x=x^(xyz), there was one other case if xyz didn't equal 1:x=1. However, z and y would both be 1 by the given equations, so xyz would still be 1.

William Ly - 2 weeks, 3 days ago

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x 1 = x x y z x^1 = x^{xyz} means that for all x x , x y z = 1 xyz=1 . For all x x means including x = 1 x=1 .

Chew-Seong Cheong - 2 weeks, 3 days ago
Razing Thunder
May 13, 2021

So this is not my own problem it was brilliant algebra quiz problem i just re-posted it answer link

0 Helpful 1 Interesting 0 Brilliant 0 Confused

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