# 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.