Prime Factorization part 2

This is an extension of prime factorization.

For any composite number, we can look at ANY factor pair that we know and derive the prime factorization.  Lets look at 36.  The prime factorization of 36 is 2 x 2 x 3 x 3.  Let's look at the picture and see the ways we can derive that from each factor pair.

Let's look at 6 x 6.  If you decompose 6 into 2 x 3, you can see how we can get to 36's prime factorization.  Now look at 9 x 4.  9 decomposes to 3 x 3 and 4 decomposes to 2 x 2.    This can be done with ANY composite number.

Now let's look at how to write prime factorization is a different way.  36's prime factorization is 2 x 2 x 3 x 3.  We can rewrite it as 2^2 x 3^2 (read as two squared times three squared OR two to the second power times three to the second power).  The picture below explains this as exponential notation.

Here is another look.  Exponential notation is on top and expanded notation is at the bottom.

One more time, but as a factor tree.

81 decomposes to 9 x 9, or 9^2 (read as nine squared or nine to the second power).  Each 9 decomposes to 3 x 3, making the prime factorization of 81 at 

3 x 3 x 3 x 3 or 3^4 (read as three to the fourth power)