### Numbers: Powers and roots

### Negative exponents

So far, we have only seen non-negative integers as an exponent in a power. We can also take negative integers as an exponent.

Negative exponents

Below we can see that when we raise the exponent by #1#, we multiply the result by #\blue2#. The other way around, if we lower the exponent by #1#, we divide by #\blue2#.

\begin{array}{rcl}\blue2^\orange0&=&1\\\blue2^\orange1&=&2\\\blue2^\orange2&=&4\\\blue2^\orange3&=&8\\\blue2^\orange4&=&16\\\blue2^\orange5&=&32\end{array}

We can define negative exponents in a similar way, every time we lower the exponent by #1#, we divide by #\blue2#.

\begin{array}{lcl}\blue2^{\orange0}&=&1\\\blue2^\orange{-1}&=&\frac{1}{2}\\\blue2^\orange{-2}&=&\frac{1}{4}\\\blue2^\orange{-3}&=&\frac{1}{8}\\\blue2^\orange{-4}&=&\frac{1}{16}\\\blue2^\orange{-5}&=&\frac{1}{32}\end{array}

We now see that #\blue2^{-\orange5}=\frac{1}{\blue2^\orange5}#. This turns out to be a general rule. We can state:

*A #\blue{\textit{base}}# raised to a negative exponent, equals #1# divided by the #\blue{\textit{base}}# raised to the corresponding positive exponent.*

**Examples**

\[\begin{array}{rcl}\blue{4}^{-\orange{3}}&=&\frac{1}{\blue4^\orange3}\\&=&\frac{1}{64} \\ \\ \blue{9}^{-\orange{6}}&=&\frac{1}{\blue{9}^\orange6}\\&=&\frac{1}{531441} \\ \\ \left(\blue{-3}\right)^{-\orange{5}}&=&\frac{1}{\left(\blue{-3}\right)^\orange5}\\&=&-\frac{1}{243} \\ \\ \left(\blue{-5}\right)^{-\orange{4}}&=&\frac{1}{\left(\blue{-5}\right)^\orange4}\\&=&\frac{1}{625}\end{array}\]

The examples we looked at so far only have integers as a base, but we can also take a fraction as our base.

Negative exponents and fractions

When we calculate #\left(\tfrac{\blue2}{\green3}\right)^{-\orange{3}}#, we find:

\[\begin{array}{rcl}\displaystyle\left(\frac{\blue2}{\green3}\right)^{-\orange{3}}&=&\displaystyle\frac{1}{\left(\frac{\blue2}{\green3}\right)^\orange3} \\&=& \displaystyle \frac{1}{\frac{\blue2}{\green3} \cdot \frac{\blue2}{\green3} \cdot {\frac{\blue2}{\green3}}} \\ &=& \displaystyle \frac{1}{\frac{8}{27}}\\&=& \displaystyle\frac{27}{8} \\ &=& \displaystyle\left(\frac{\green3}{\blue2}\right)^\orange3 \end{array}\]

In general, we can therefore say:

*A fraction raised to a negative exponent equals the reciprocal of the fraction raised to the corresponding positive exponent.*

**Examples**

\[\begin{array}{rcl}\left(\frac{\blue3}{\green4}\right)^{-\orange4}&=&\left(\frac{\green4}{\blue3}\right)^\orange4\\&=&\frac{256}{81} \\ \\ \left(\frac{\blue5}{\green2}\right)^{-\orange2}&=&\left(\frac{\green2}{\blue5}\right)^\orange2\\&=&\frac{4}{25} \\ \\ \left(\frac{\blue4}{\green7}\right)^{-\orange3}&=&\left(\frac{\green7}{\blue4}\right)^\orange3\\&=&\frac{343}{64} \\ \\ \left(-\frac{\blue4}{\green3}\right)^{-\orange5}&=&\left(-\frac{\green3}{\blue4}\right)^\orange5\\&=&-\frac{243}{1024} \end{array}\]

#\begin{array}{rcl}

6^{-3} &=&\dfrac{1}{6^{3}}\\ &&\phantom{xxx}\blue{\text{calculation rule: a base raised to a negative exponent equals }} \\ &&\phantom{xxx}\blue{1 \text{ divided by the base raised to the corresponding positive exponent}}\\

&=& \displaystyle {{1}\over{216}} \\ &&\phantom{xxx}\blue{\text{calculated}}

\end{array}#

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