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How do I prove that for all natural numbers $n$ and complex numbers $a, b, c, z, w$ if $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$.

Thanks.

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  • $\begingroup$ What is the question? $\endgroup$ – blub Apr 1 at 20:33
  • $\begingroup$ Perhaps you mean to say "Prove or disprove that if $a$ is an $n^{\textrm{th}}$ root of $z$ and $b$ is an $n^{\textrm{th}}$ root of $w$, then $ab$ is an $n^{\textrm{th}}$ root of $zw$"? $\endgroup$ – MPW Apr 1 at 20:35
  • $\begingroup$ @blub I just added it sorry. $\endgroup$ – Sania Apr 1 at 20:35
  • $\begingroup$ @Sania "Prove or disprove" is not a question, it's a command. $\endgroup$ – ajotatxe Apr 1 at 20:37
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You have $a^n=z$ , $b^n=w$ and $c^n=zw$ so $\left(\frac{ab}c\right)^n=1\quad$ (the case $z=0$ or $w=0$ being trivial).

Thus all we can deduce is $\frac{ab}c$ is a n-root of unity, but not necessarily $1$.

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A trivial counter example would be $n=2, z=w=a=b=1$ and $c=-1$.

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