How To Teach Math - Exponent Laws

A1.1.4 Use the laws of exponents for rational exponents

Resources

Topic Discussion

This section deals with rational exponents. For a discussion of multiplying/dividing and powers of exponents check out the polynomial section on the left. For this section we are going to talk about non-integer rational exponents. This is often a difficult topic for students to grasp. The first goal I usually have is to first convince students that a number to the one-half power is equivalent to taking the square root of that number. I use the following method to accomplish this task.

I introduce the topic of simplifying square roots by trying to establish a pattern. I start with asking what is the square root of 4 squared. Then I continue to use even powers until they realize that you divide the exponent by 2. During the square root lesson I usually teach the students to subtract one from the exponent if it is odd and leave it inside the square root. That is the proper way to simplify square roots. But before we get there I allow the discussion to go to a place that can help us with fractional exponents. The conversation might go like this:

Me: So we know the square root of "anything" to an even power is that "anything" to half the original power. So what do we think is the square root of x to the 3rd power?

Student: Square root of x to the 1.5 power?

Me: Exactly!!! So what do you think the square root of x is?

Student: Uhh...

Me: Well what is the exponent of x?

Student: 1...oh so the the square root of x would x to the .5.

Me: Yes! What is another way to say .5?

Student: One-half.

Me: Great...so we just showed that the square root of anything is that anything to the one-half power. However, for our purposes now...

I have had this exact conversation in a Pre-algebra classroom before. Now when I get to rational exponents I just have to recall this conversation and I am now ready to talk about fractional exponents. It is even appropriate to have this conversation for the sake of the students that missed it the first time around. It is then very natural to extend this conversation to other powers with denominator 2. At this point you can decide if you want to use this to generalize the rules of fractional exponents. The other option is to reinforce this learning by considering the same pattern using cube roots instead of square roots.