Exponential Properties Involving Quotients

Key Questions

  • What is the quotient of powers property?

    #(a^m)/(a^n)=a^(m-n)#

    This property allows you to simplify problems where you have a fraction of the same numbers ( #a#) raised to different powers (#m and n#).
    For example:

    #(3^3)/(3^2)=(3*3*3)/(3*3)=3^(3-2)=3#

    You can see how the power of 3, in the numerator, is "reduced" by the presence of the power 2 in the denominator.

    You can also check te result by doing the multiplications:

    #(3^3)/(3^2)=(3*3*3)/(3*3)=27/9=3#

    As a challenge try to find out what happens when #m=n# !!!!!

    Gió · 5 · Dec 19 2014
  • What is the power of a quotient property?

    The Power of a Quotient Rule states that the power of a quotient is equal to the quotient obtained when the numerator and denominator are each raised to the indicated power separately, before the division is performed.
    i.e.: #(a/b)^n=a^n/b^n#
    For example:
    #(3/2)^2=3^2/2^2=9/4#

    You can test this rule by using numbers that are easy to manipulate:
    Consider: #4/2# (ok it is equal to #2# but for the moment let it stay as a fraction), and let us calculate it with our rule first:
    #(4/2)^2=4^2/2^2=16/4=4#
    Let us, now, solve the fraction first and then raise to the power of #2#:
    #(4/2)^2=(2)^2=4#

    This rule is particularly useful if you have more difficult problems such as an algebraic expression (with letters):
    Consider: #((x+1)/(4x))^2#
    You can now write:
    #((x+1)/(4x))^2=(x+1)^2/(4x)^2=(x^2+2x+1)/(16x^2)#

    Gió · 3 · Dec 25 2014
  • How do you simplify expressions using the quotient rule?

    The Quotient Rule for Exponents

    Let me give you a basic explanation:

    Lets take the example of
    #4^36/4^21#

    The quotient rule states that for an expression like #x^a/x^b = x^(a-b)#

    Now of course you question how to simplify expressions using this rule.

    Now lets take such a eg.

    Compute the following: #{625x^23}/{25x^3}#

    this nothing but 25#(x^(23-3))#

    So we are left with this final answer #25x^20#

    Sidharth · 1 · Jan 24 2015

Questions

Exponents and Exponential Functions