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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