Why Expressions with Negative Exponents Equal their Reciprocals

We are familiar with the rule that for a positive exponent m,

x^{- m} = \displaystyle\frac{1}{x^m}

and

\displaystyle\frac{1}{x^{-m}} = x^m.


In this post, we learn the reason behind the concept of negative exponents and their relationship to the reciprocal of the algebraic expression containing them.

Recall that in dividing an algebraic expression with the same base, we have to subtract their exponents. For example, for m > n

\displaystyle\frac{x^m}{x^n} = x^{m-n}.

Using this law, it follows that  x^{-m} = x^{0-m}.  But

x^{0-m} = \displaystyle\frac{x^0}{x^m}.

Since x^0 = 1,

\displaystyle\frac {x^0}{x^m} = \frac{1}{x^m}.

Therefore,

x^{-m} = \displaystyle\frac{1}{x^m}

The second equation above is a consequence of this result.

\displaystyle\frac{1}{x^{-m}} = \frac{1}{\frac{1}{x^m}} = x^m.

Those are the reasons why expressions with negative exponents equal their reciprocals.

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3 thoughts on “Why Expressions with Negative Exponents Equal their Reciprocals

  1. Hi Guillermo,
    Just wanted to let you know that I have finally gotten around to completing my post on this topic on my website, and I gave you a mention as well in the post. Please let me know if you have any issues with what I’ve put up.
    Cheers,
    Shaun

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