Rational exponents and geometric sequences

The CME curriculum has a really clever way to help kids gain another perspective on rational exponents. What’s so confusing about rational exponents — I think — is that it’s very difficult to make sense of them using the model of exponentiation as repeated multiplication. How do you multiply something half a time?

CME subverts this by actually talking about multiplication of something a fractional number of times.

What follows is a direct quote from the CME textbook:

You can use a geometric sequence to list the integer powers of any  number by starting with 1 and repeatedly multiplying by that number. Here are the powers of 27: 1; 27, 729; 19683…

If you multiply 1 by 27 four times, or multiply 4 factors of 27, you get 531,441. So 27^4 = 531,441. But what is 27^{\frac{2}{3}}? How can you multiply 1 by something \frac{2}{3} of one time?

One way is to insert extra terms in the geometric sequence between 1 and 27:

1, __, __, 27, __, __, 729, ___, …

Assume it remains a geometric sequence. What goes in the blanks?

As far as turning this into a problem that kids could solve, I’m imagining something like giving them a set of problems that includes both the sort of incomplete sequences that were shown above, and also a request to imagine what it might mean to multiply 1 by 16 a half of one time. But if you’ve got a different idea on how to turn this into a good problem, drop into the comments.

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One thought on “Rational exponents and geometric sequences

  1. Ooooh, I like this. Fractional exponents is in Algebra CCSS, so I can use it this year. Develop it this way first, then you can go back and discuss ow it fits with properties of exponents. Could add in some sort of context – bacteria growth or money?? I like it with just trying to fill in the patterns too – make a table with n=0,1,2,3 and “nice” outputs, then “expand the table to n=0,1/2,1,3/2,2 or n=0,1/3,2/3,1,4/3, etc. depending on the value you chose?

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