What are Negative Exponents?

1 Answer
Feb 14, 2015

Negative exponents are an extension of the initial exponent concept.

To understand negative exponents ,
first review what we mean by positive (integer) exponents

What do we mean when we write something like:
#n^p# (for now, assume that #p# is a positive integer.

One definition would be that
#n^p# is #1# multiplied by #n#, #p# times.

Note that using this definition
#n^0# is #1# multiplied by #n#, #0# times
i.e. #n^0 = 1# (for any value of #n#)

Suppose you know the value of #n^p# for some particular values of #n# and #p#
but you would like to know the value of #n^q# for a value #q# less than #p#

For example suppose you knew that
#2^10 = 1024# but you wanted to know what #2^9# was equal to.
Is there a faster way than multiplying #1# by #2#, #9# times?
Yes.
If we note that #2^9 = (2^10)/2#
we can simply divide #1024# by #2# (giving 512) to obtain #2^9#

In general if we know that the value of #n^p# is #k#
and we want to know the value of #n^q# when #q<p#
we can simply divide k by n^(p-q)

With this in mind what is the value of
#n^(-t)# ?
We know that #n^0 = 1#
so #n^(-t)# must be #1# divided by #n#, #(0 - (-t))# times

That is #n^(-t) = 1/n^t#

As a final example consider the descending powers of 3 in the following, noting that with each line down the result is decreased by dividing the current value by 3

#3^4 = 81#
#3^3 = 27#
#3^2 = 9#
#3^1 = 3#
#3^0 = 1#
#3^(-1) = 1/3#
#3^(-2) = 1/9#
#3^(-3) = 1/27#