In my last post, you saw another maths shortcut – exponents. But I used examples where the exponents were positive integers. It turns out that mathematically, any number, be it negative, fractional, or irrational, can be used as an exponent. Let’s expand our knowledge here by looking at exponents that are not positive integers.

In my last post, I presented the rule that

\[\frac{{x}^{a}}{{x}^{b}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{a}{-}{b}}

\]

so that

\[

\frac{{x}^{3}}{{x}^{2}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{3}{-}{2}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{1}\hspace{0.33em}{=}\hspace{0.33em}{x}

\].

What if you have \[

\frac{{x}^{3}}{{x}^{3}}

\]?

Well, the rule says that

\[\frac{{x}^{3}}{{x}^{3}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{3}{-}{3}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{0}

\]

But we know that the same number divided by itself is 1. So it makes sense, and it makes maths consistent, if we define anything raised to the “0” power as 1. That is,

\[

{x}^{0}\hspace{0.33em}{=}\hspace{0.33em}{1}

\] for any number *x*.

Now let’s go further. What if we have \[

\frac{{x}^{2}}{{x}^{3}}

\]? According to the rule, this is \[

{x}^{{2}{-}{3}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{-}{1}}

\]. What does a negative exponent mean? Well let’s do the same problem without using the rule:

\frac{{x}^{2}}{{x}^{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{x}}{\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}{x}}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{{x}^{1}}

\]

So it appears that a negative exponent means that the number raised to a negative power, is the same as the number in the denominator raised to its positive power. This is true for any exponent:

\[{x}^{{-}{a}}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{{x}^{a}}

\]

This means that factors in a fraction can be moved at will from the numerator to the denominator or vice versa, by just changing the sign of the exponents:

\[\begin{array}{l}

{\frac{7}{{x}^{{-}{2}}}\hspace{0.33em}{=}\hspace{0.33em}{7}{x}^{2}}\\

{{3}{y}^{{-}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{{y}^{3}}}\\

{\frac{4{xy}^{5}}{{wz}^{{-}{6}}}\hspace{0.33em}{=}\hspace{0.33em}\frac{4{xy}^{5}{z}^{6}}{w}}

\end{array}

\]

However, be careful. You can only do this with factors, that is things that are multiplied together. It does not work with fractions where things are added or subtracted:

\[\frac{x}{{y}\hspace{0.33em}{+}\hspace{0.33em}{z}^{{-}{3}}}\hspace{0.33em}\ne\hspace{0.33em}\frac{{x}\hspace{0.33em}{+}\hspace{0.33em}{z}^{3}}{y}

\]

By the way, the symbol ≠ means “does not equal”.