Negative Exponents

Key Questions

  • What is the exponent of zero property?

    I suppose you mean the fact that a number to the zero exponent is always equal to one, for example:

    #3^0=1#

    The intuitive explanation can be found remembering that:
    1) dividing two equal numbers gives 1;
    ex. #4/4=1#
    2) The fraction of two equal numbers a to the power of m and n gives:
    #a^m/a^n=a^(m-n)#

    Now:
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    Gió · 1 · Dec 14 2014
  • What are Negative Exponents?

    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#

    Alan P. · 1 · Feb 14 2015
  • How do negative exponents affect fractions?

    Raising to the -1 power is equivalent to taking the reciprocal, so we have

    #(a/b)^{-1}=b/a#

    I hope that this was helpful.

    Wataru · 2 · Dec 5 2014
  • How do you simplify expressions with negative exponents?

    #x^(-n) = 1/(x^n)#

    Maybe you were asking for something more than this (???)

    Alan P. · 1 · Feb 28 2015
  • How do you evaluate expressions with negative exponents?

    You can start by rewriting in the following way:

    #b^{-x}=1/b^x#

    I hope that this was helpful.

    Wataru · 1 · Nov 6 2014

Questions

Exponents and Exponential Functions