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Recently I have been chewing on a bit of a paradox in my mind, and I'm trying to figure out what I'm doing wrong. I am not in high school anymore, I have just graduated from college, and I haven't actively done any non-CompSci related math in a long time aside from youtube videos. So, I'm sorry if this question is really basic, I am versed in this kind of stuff, I am just having problems googling the answer to this one.

Alright, so back in middle school Algebra we learned $(b^n)^m = b^{nm}$. What I'm wondering is, say b is a negative number raised to an odd power. Couldn't we write it in such a way that the answer is always positive?

$b^n$ where $b$ is negative and $n$ is odd. Can't we always write it in the form $(b^2)^{n/2}$. Following the order of operations we can evaluate the parenthesis first, get a positive number, and from there it's a positive number raised to a positive, rational exponent. Which is always positive. Yet, if you did the expansion of the exponents, the answer is clearly negative.

My googling of this property of exponents hasn't led to anywhere that stipulates that this property is only valid when $b > 0$, and from my point of view it's a very trivial question, so what am I doing wrong here?


marked as duplicate by hardmath, Leucippus, Namaste algebra-precalculus Jan 5 '18 at 1:03

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  • $\begingroup$ When you take the square root of anything, you get a positive and a negative root. You took the negative root, squared it and then took the positive square root. It will be alright if you take the negative root for your question. $\endgroup$ – Mohammad Zuhair Khan Jan 4 '18 at 16:41
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    $\begingroup$ Raising a negative real number to a fractional power introduces ambiguity. The Law of Exponents you refer to is only valid when the base $b$ is a positive real number or the exponents $m,n$ are integers. $\endgroup$ – hardmath Jan 4 '18 at 16:43
  • $\begingroup$ The exponent notation should be used only with positive numbers,'except in the case of an integer exponent, for consistency reasons. $\endgroup$ – Bernard Jan 4 '18 at 16:45

Generally speaking, the powers of negative numbers raise issues, such as the power law

$$b^{mn}=(b^m)^n$$ being invalid.

As a simple example,

$$(-1)^1=((-1)^2)^{1/2}$$ does not hold.

  • $\begingroup$ I think you are only repeating a particular instance of the invalid "expansion" that the OP is asking about. Note the final paragraph, "My googling of this property of exponents hasn't led to anywhere that stipulates that this property is only valid when $b\gt 0$." So an Answer would show the OP such an explanation, although in my opinion this is an often duplicated topic. $\endgroup$ – hardmath Jan 4 '18 at 16:53
  • $\begingroup$ @hardmath: a simple way to show that the law doesn't hold is by giving a counterexample, which I did. I am confirming that what the OP thinks is... wrong. $\endgroup$ – Yves Daoust Jan 4 '18 at 17:01

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