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## Main Question or Discussion Point

What's actually the definition of [itex]a^{m/n}[/itex] where m and n are integers and a is any real number? Suppose I define it as the n-th square root of [itex]a^m[/itex]. Wouldn't it be inconsistent with other stuffs?

What stuffs? For example, [itex]a^1[/itex] is supposed to be a. But 1 = 2/2 and, using my earlier definition, [itex]a^{2/2}=\sqrt{a^2} = |a|[/itex]. Thus if we use my definition, [itex]a^1[/itex] wouldn't be the same as [itex]a^{2/2}[/itex] for a < 0.

So, what's the definition in use for [itex]a^{m/n}[/itex]?

Thanks a lot.

What stuffs? For example, [itex]a^1[/itex] is supposed to be a. But 1 = 2/2 and, using my earlier definition, [itex]a^{2/2}=\sqrt{a^2} = |a|[/itex]. Thus if we use my definition, [itex]a^1[/itex] wouldn't be the same as [itex]a^{2/2}[/itex] for a < 0.

So, what's the definition in use for [itex]a^{m/n}[/itex]?

Thanks a lot.