One of my students, Sarah, was really confused by the process of simplifying complicated expressions containing many different variables raised to different powers, such as: (a^2 * b^-4 * c^5)/(a * b^-2 * c^-3). I decided that the best way to help her was to give her example with numerals instead of variables, to provide the scoffolding necessary for her to understand the more abstract principles at work.
I asked Sarah to consider the multiplication of 3^2 and 3^3. If we write 3^2 as 3 * 3 and 3^3 as 3 * 3 * 3, then we can write 3^2 * 3^3 as 3 * 3 * 3 * 3 * 3. Sarah was quickly able to tell me that this can be written as 3^5, and this is why 3^2 * 3^3 is the same as 3^(2+3).
I then asked Sarah to consider 3^5/3^2. I wrote this out as (3 * 3 * 3 * 3 * 3)/(3 * 3). I asked her to simplify this expression. She canceled two of the 3's in the numerator and denominator, leaving 3 * 3 * 3. This allowed her to see why 3^5/3^2 = 3^(5-2). At this point, she was comfortable working with variables. She quickly picked up the ability to simplify expressions such as x^4/x^2. With a little more direction, she was able to conquer the more complicated expressions as well, and had a greater understanding of negative exponents as well as positive ones.
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