Lesson WHY negative exponents are defined the way they are

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Negative numbers were invented by mathematicians so that operation of subtraction -- subtracting one number from another -- would be defined even if the number that we subtract is bigger than the number that we are subtracting from.

Negative exponents were invented with the same goal in mind, to allow operation of "power" to be used with more kinds of numbers. In this case it would be taking a number to the power of a negative. Example: {{{3^-2}}}.

Mathematicians were free to define this operation however they wanted. They decided to define it so that the rule of multiplying exponents is preserved:

{{{highlight( a^x*a^y = a^(x+y) )}}}.

So, let's now consider a simple example: how would you choose what value should be {{{3^-2}}}, so that it makes sense? It is very simple. Multiply {{{3^-2}}} by {{{3^2}}}, and assume that the rule of multiplying exponents is preserved:

{{{3^-2*3^2 = 3^(-2+2) = 3^0 = 1}}}

rewriting more simply:

{{{3^-2*3^2 = 1}}}

Now, divide both sides by 3^2:

{{{3^-2 = 1/3^2 }}}

From here, it is clear that if we want to define the operation of power with a negative exponeent, it should be the <A HREF=Multiplicative-inverse.wikipedia>multiplicative inverse</A> of the same operation with the positive power.

We could easily use any numbers a, b in my example above to get: {{{ highlight( a^(-b) = 1/a^b ) }}}.

Some examples:

{{{2^-2 = 1/2^2 = 1/4}}}

{{{ 3^-3 = 1/3^3 = 1/27 }}}

{{{5^-1 = 1/5^1 = 1/5}}}