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

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