The division properties of exponents are: when dealing with like bases, exponents are subtracted when the bases are divided; when an entire quotient is raised to an exponential power, both the numerator and denominator are raised to the power before division is performed. One way to employ the division properties of exponents is to expand the terms above and below the dividing line for like bases.
For instance, a^n divided by a^m equals a^n-m. Using numbers for variables, 2^4 divided by 2^2 power is 16 divided by 4, which equals 4. Written in exponential form, 2^2 equals 4. For an entire quotient that has an exponent, an example is (2/4)^3. Take 2^3 divided by 4^3, which comes out to 8/64 and may be reduced to 1/8.
The long way to do the math is to write out the individual parts in long form. The first example is 2*2*2*2 divided by 2*2. Two of the numerals on top cancel out the two numerals on the bottom. That leaves 2*2 which is 4.
Numbers in front of bases are also divided in the quotient. A formula 6b^4 divided by 3b^2 breaks down to 2b^2. When the exponent of a numerator is greater than the denominator, the fraction is inverted or the exponent is a negative number. For instance, a^4 divided by a^6 is a^-2 or 1/a^2.