Thoughts On The Distributive Property

If you are a math teacher, you teach or use the distributive property at least once during the year but sometimes more.  Today while reviewing the properties of exponents, I realized I could apply the idea of distribution to one of the exponent rules.

If its in algebra we have something like 3(X + 2) or if its elementary we might use 10(3 + 2).  We multiply the outside number or varible with each term inside the parenthesis.  Pretty standard stuff except about 1/3rd of the students forget the second term so the first might end up as 3x + 2 instead of 3x + 6.   As for the other, if you have students who remember PEMDAS, they are likely to ask why we don't add 3 + 2 first since that is what you are supposed to do.   That is a discussion for another day.

I also use the distributive method as one of the ways I teach multiplication of binomials.

I have students split the first binomial so the 2x is on the left and 3 is on the right.  I keep the second binomial whole place one by the 2x and the again by the 3 as in the photo to the right.
Multiply, then combine common terms and you have an answer.

I've found several of my students actually prefer this method to the foil because they find it less confusing.

Today I reviewed the rules for exponents.  You know the ones we all know by heart. If you are multiplying terms with exponents and they have the same base you add exponents. 

X^2 * x^3 * x^2 = x^7

The one my students have trouble is the power to a power where you multiply exponents.  They get it if there is one variable such as (x^2)^5, they know to multiply the exponents and get x^10.  They run into problems if its a bit more extensive such as (2x^2y^5)^5

I realized today the processes is similar to the distributive property because you are multiplying the exponents of the terms inside the parenthesis by the exponent outside.

If you see the drawing, I've removed the terms and only looked at the exponents to show my students how you multiply each one.

We are distributing the outside across the inside so why can't we include this property to reinforce the process?  I did see a few light bulbs go off in the crowd because they could relate the process to prior knowledge.

I am always looking for ways to activate prior knowledge and this seemed like a good way to do it.

I would love feedback on how you all see this as a way to teach this topic?  Do you think its good or am I way off base?  Thanks for your comments ahead of time.