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I recently was working with square roots and came across this-

$({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1$

I understand that this is not true,but despite repeated attempts failed to prove it wrong.Can someone please point out the fallacy in this proof.Thanks in advance.

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    $\begingroup$ You could simplify your fallacy: $-1=(-1)^{1}=(-1)^{\frac{2}{2}}=((-1)^2)^{\frac{1}{2}}=1^{\frac{1}{2}}=1$. ;-) $\endgroup$ – Blackbird Aug 13 '15 at 9:26
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The property that $(a^n)^m=(a^m)^n=a^{mn}$ is only defined for real numbers. But $\sqrt{-1}=i$ is a complex number.

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    $\begingroup$ See my comment above. ;-) $\endgroup$ – Blackbird Aug 13 '15 at 9:27
  • $\begingroup$ Only for real positive numbers if one wants the property to be true for all $m,n\in\mathbb{R}$. $\endgroup$ – A.Γ. Aug 13 '15 at 11:58
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The fourth root of $1$ has four solutions: $1$, $-1$, $j$ and $-j$. The last two solutions are identical to the two solutions of the square root of $-1$.

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  • $\begingroup$ What is $j$? Are you talking about $\iota$? $\endgroup$ – Aditya Agarwal Aug 13 '15 at 9:23
  • $\begingroup$ Yes, exactly. Different notation, apparently. $\endgroup$ – Bernhard Aug 13 '15 at 9:27
  • $\begingroup$ @Bernhard-Last two solutions may be identical but the first two are also solutions...why are they not acceptable? $\endgroup$ – tatan Aug 28 '15 at 14:53
  • $\begingroup$ The reason why the first two solutions are not correct is given by the comment of Aditya Agarwal: The product rule for powers is only valid for positive real numbers. $\endgroup$ – Bernhard Aug 29 '15 at 18:23

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